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CalculusCalculus (MindTap Course List)(a) Use (4) to show that if s n is the nth partial sum of the harmonic series, then s n ≤ 1 + ln n (b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22.BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 11.3, Problem 43E

Textbook Problem

(a) Use (4) to show that if

(b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22.

Calculus (MindTap Course List)

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Ch. 11.1 - a What is a sequence? b What does it mean to say...Ch. 11.1 - a What is a convergent sequence? Give two...Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - List the first five terms of the sequence....

Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - List the first five terms of the sequence....Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Find a formula for the general term an of the...Ch. 11.1 - Calculate, to four decimal places, the first ten...Ch. 11.1 - Calculate, to four decimal places, the first ten...Ch. 11.1 - Calculate, to four decimal places, the first ten...Ch. 11.1 - Calculate, to four decimal places, the first ten...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Determine whether the sequence converges or...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - Use a graph of the sequence to decide whether the...Ch. 11.1 - a Determine whether the sequence defined as...Ch. 11.1 - If 1000 is invested at 6 interest, compounded...Ch. 11.1 - If you deposit 100 at the end of every month into...Ch. 11.1 - A fish farmer has 5000 catfish in his pond. The...Ch. 11.1 - Find the first 40 terms of the sequence defined...Ch. 11.1 - For what values of r is the sequence {nrn}...Ch. 11.1 - a If an is convergent, show that limnan+1=limnan b...Ch. 11.1 - Suppose you know that {an} is a decreasing...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Determine whether the sequence is increasing,...Ch. 11.1 - Find the limit of the sequence {2,22,222,...}Ch. 11.1 - A sequence an is given by a1=2,an+1=2+an. a By...Ch. 11.1 - Show that the sequence defined by a1=1an+1=31an is...Ch. 11.1 - Show that the sequence defined by a1=2an+1=13an...Ch. 11.1 - a Fibonacci posed the following problem: Suppose...Ch. 11.1 - a Let a1=a,a2=f(a),a3=f(a2)=f(f(a)),...,...Ch. 11.1 - a Use a graph to guess the value of the limit...Ch. 11.1 - Use Definition 2 directly to prove that limnrn=0...Ch. 11.1 - Prove Theorem 6. Hint: Use either Definition 2 or...Ch. 11.1 - Prove Theorem 7.Ch. 11.1 - Prove that if limnan=0 and {bn} is bounded, then...Ch. 11.1 - Let an=(1+1n)n. a Show that if 0ab, then...Ch. 11.1 - Let a and b be positive numbers with ab. Let a1 be...Ch. 11.1 - a Show that if limna2n=L and limna2n+1=L, then...Ch. 11.1 - The size of an undisturbed fish population has...Ch. 11.2 - a What is the difference between a sequence and a...Ch. 11.2 - Explain what it means to say that n=1an=5.Ch. 11.2 - Calculate the sum of the series n=1an whose...Ch. 11.2 - Calculate the sum of the series n=1an whose...Ch. 11.2 - Calculate the first eight terms of the sequence of...Ch. 11.2 - Calculate the first eight terms of the sequence of...Ch. 11.2 - Calculate the first eight terms of the sequence of...Ch. 11.2 - Calculate the first eight terms of the sequence of...Ch. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Find at least 10 partial sums of the series. Graph...Ch. 11.2 - Let an=2n3n+1. a Determine whether {an} is...Ch. 11.2 - a Explain the difference between i=1nai and j=1naj...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the geometric series is...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Determine whether the series is convergent or...Ch. 11.2 - Let x=0.99999.... a Do you think that x1 or x=1? b...Ch. 11.2 - A sequence of terms is defined by a1=1 an=(5n)an1...Ch. 11.2 - Express the number as a ratio of integers....Ch. 11.2 - Express the number as a ratio of integers....Ch. 11.2 - Express the number as a ratio of integers....Ch. 11.2 - Express the number as a ratio of integers....Ch. 11.2 - Express the number as a ratio of integers....Ch. 11.2 - Express the number as a ratio of integers. 5.71358Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - Find the values of x for which the series...Ch. 11.2 - We have seen that the harmonic series is a...Ch. 11.2 - Use the partial fraction command on your CAS to...Ch. 11.2 - Use the partial fraction command on your CAS to...Ch. 11.2 - If the nth partial sum of a series n=1an is...Ch. 11.2 - If the nth partial sum of a series n=1an is...Ch. 11.2 - A doctor prescribes a 100-mg antibiotic tablet to...Ch. 11.2 - A patient is injected with a drug every 12 hours....Ch. 11.2 - A patient takes 150 mg of a drug at the same time...Ch. 11.2 - After injection of a dose D of insulin, the...Ch. 11.2 - When money is spent on goods and services, those...Ch. 11.2 - A certain ball has the property that each time it...Ch. 11.2 - Find the value of c if n=2(1+c)n=2Ch. 11.2 - Find the value of c such that n=0enc=10Ch. 11.2 - In Example 9 we showed that the harmonic series is...Ch. 11.2 - Graph the curves y=xn, 0x1, for n=0,1,2,3,4,... on...Ch. 11.2 - The figure shows two circles C and D of radius 1...Ch. 11.2 - A right triangle ABC is given with A= and |AC|=b....Ch. 11.2 - What is wrong with the following calculation?...Ch. 11.2 - Suppose that n=1(an0) is known to be a convergent...Ch. 11.2 - Prove part i of Theorem 8.Ch. 11.2 - If an is divergent and c0 is divergent, show that...Ch. 11.2 - If an is convergent and bn is divergent, show that...Ch. 11.2 - If an and bn are both divergent, is (an+bn)...Ch. 11.2 - Suppose that a series an has positive terms and...Ch. 11.2 - The Fibonacci sequence was defined in Section 11.1...Ch. 11.2 - The Cantor set, named after the German...Ch. 11.2 - a A sequence {an} is defined recursively by the...Ch. 11.2 - Consider the series n=1nn+1!. a Find the partial...Ch. 11.2 - In the figure at the right there are infinitely...Ch. 11.3 - Draw a picture to show that n=21n1.311x1.3dx What...Ch. 11.3 - Suppose f is a continuous positive decreasing...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Use the Integral Test to determine whether the...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Determine whether the series is convergent or...Ch. 11.3 - Explain why the Integral Test cant be used to...Ch. 11.3 - Explain why the Integral Test cant be used to...Ch. 11.3 - Find the values of p for which the series is...Ch. 11.3 - Find the values of p for which the series is...Ch. 11.3 - Find the values of p for which the series is...Ch. 11.3 - Find the values of p for which the series is...Ch. 11.3 - The Riemann zeta-function is defined by...Ch. 11.3 - Leonhard Euler was able to calculate the exact sum...Ch. 11.3 - Euler also found the sum of the p-series with p=4:...Ch. 11.3 - a Find the partial sum s10 of the series n11/n4....Ch. 11.3 - a Use the sum of the first 10 terms to estimate...Ch. 11.3 - Find the sum of the series n=1ne2n correct to four...Ch. 11.3 - Estimate n=1(2n+1)6 correct to five decimal...Ch. 11.3 - How many terms of the series n=21/[n(lnn)2] would...Ch. 11.3 - Show that if we want to approximate the sum of the...Ch. 11.3 - a Show that the series n1(lnn)2/n2 is convergent....Ch. 11.3 - a Use 4 to show that if sn is the nth partial sum...Ch. 11.3 - Use the following steps to show that the sequence...Ch. 11.3 - Find all positive values of b for which the series...Ch. 11.3 - Find all values of c for which the following...Ch. 11.4 - Suppose an and bn are series with positive terms...Ch. 11.4 - Suppose an and bn are series with positive terms...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Determine whether the series converges or...Ch. 11.4 - Use the sum of the first 10 terms to approximate...Ch. 11.4 - Use the sum of the first 10 terms to approximate...Ch. 11.4 - Use the sum of the first 10 terms to approximate...Ch. 11.4 - Use the sum of the first 10 terms to approximate...Ch. 11.4 - The meaning of the decimal representation of a...Ch. 11.4 - For what values of p does the series n=21/(nplnn)...Ch. 11.4 - Prove that if an0 and an converges, then an2 also...Ch. 11.4 - a Suppose that an and bn are series with positive...Ch. 11.4 - a Suppose that an and bn are series with positive...Ch. 11.4 - Give an example of a pair of series an and bn with...Ch. 11.4 - Show that if an0 and limnnan0, then an is...Ch. 11.4 - Show that if an0 and an is convergent, then...Ch. 11.4 - If an is a convergent series with positive terms,...Ch. 11.4 - If an and bn are both convergent series with...Ch. 11.5 - a What is an alternating series? b Under what...Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Test the series for convergence or divergence....Ch. 11.5 - Graph both the sequence of terms and the sequence...Ch. 11.5 - Graph both the sequence of terms and the sequence...Ch. 11.5 - Show that the series is convergent. How many terms...Ch. 11.5 - Show that the series is convergent. How many terms...Ch. 11.5 - Show that the series is convergent. How many terms...Ch. 11.5 - Show that the series is convergent. How many terms...Ch. 11.5 - Approximate the sum of the series correct to four...Ch. 11.5 - Approximate the sum of the series correct to four...Ch. 11.5 - Approximate the sum of the series correct to four...Ch. 11.5 - Approximate the sum of the series correct to four...Ch. 11.5 - Is the 50th partial sum s50 of the alternating...Ch. 11.5 - For what values of p is each series convergent?...Ch. 11.5 - For what values of p is each series convergent?...Ch. 11.5 - For what values of p is each series convergent?...Ch. 11.5 - Show that the series (1)n1bn, where bn=1/n if n is...Ch. 11.5 - Use the following steps to show that n=1(1)n1n=ln2...Ch. 11.6 - What can you say about the series an in each of...Ch. 11.6 - Determine whether the series is absolutely...Ch. 11.6 - Determine whether the series is absolutely...Ch. 11.6 - Determine whether the series is absolutely...Ch. 11.6 - Determine whether the series is absolutely...Ch. 11.6 - Determine whether the series is absolutely...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Ratio Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use the Root Test to determine whether the series...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - Use any test to determine whether the series is...Ch. 11.6 - The terms of a series are defined recursively by...Ch. 11.6 - A series an is defined by the equations...Ch. 11.6 - Let {bn} be a sequence of positive numbers that...Ch. 11.6 - Let {bn} be a sequence of positive numbers that...Ch. 11.6 - For which of the following series is the Ratio...Ch. 11.6 - For which positive integers k is the following...Ch. 11.6 - a Show that n=0xn/n! converges for all x. b Deduce...Ch. 11.6 - Let an be a series with positive terms and let...Ch. 11.6 - a Find the partial sum s5 of the series...Ch. 11.6 - Use the sum of the first 10 terms to approximate...Ch. 11.6 - Prove the Root Test. Hint for part i: Take any...Ch. 11.6 - Around 1910, the Indian mathematician Srinivasa...Ch. 11.6 - Given any series an, we define a series an+ whose...Ch. 11.6 - Prove that if an is a conditionally convergent...Ch. 11.6 - Suppose the series an is conditionally convergent....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.7 - Test the series for convergence or divergence....Ch. 11.8 - What is a power series?Ch. 11.8 - a What is the radius of convergence of a power...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - Find the radius of convergence and interval of...Ch. 11.8 - If n=0cn4n is convergent, can we conclude that...Ch. 11.8 - Suppose that n=0cnxn converges when x=4 and...Ch. 11.8 - If k is a positive integer, find the radius of...Ch. 11.8 - Let p and q be real numbers with pq. Find a power...Ch. 11.8 - Is it possible to find a power series whose...Ch. 11.8 - Graph the first several partial sums sn(x) of the...Ch. 11.8 - The function J1 defined by...Ch. 11.8 - The function A defined by...Ch. 11.8 - A function f is defined by f(x)=1+2x+x2+2x3+x4+...Ch. 11.8 - If f(x)=n=0cnxn, where cn+4=cn for all n0, find...Ch. 11.8 - Show that if limn|cn|n=c where c0, then the radius...Ch. 11.8 - Suppose that the power series cn(xa)n satisfies...Ch. 11.8 - Suppose the series cnxn has radius of convergence...Ch. 11.8 - Suppose that the radius of convergence of the...Ch. 11.9 - If the radius of convergence of the power series...Ch. 11.9 - Suppose you know that the series n=0bnxn converges...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Express the function as the sum of a power series...Ch. 11.9 - Express the function as the sum of a power series...Ch. 11.9 - a Use differentiation to find a power series...Ch. 11.9 - a Use Equation 1 to find a power series...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for the...Ch. 11.9 - Find a power series representation for f, and...Ch. 11.9 - Find a power series representation for f, and...Ch. 11.9 - Find a power series representation for f, and...Ch. 11.9 - Find a power series representation for f, and...Ch. 11.9 - Evaluate the indefinite integral as a power...Ch. 11.9 - Evaluate the indefinite integral as a power...Ch. 11.9 - Evaluate the indefinite integral as a power...Ch. 11.9 - Evaluate the indefinite integral as a power...Ch. 11.9 - Use a power series to approximate the definite...Ch. 11.9 - Use a power series to approximate the definite...Ch. 11.9 - Use a power series to approximate the definite...Ch. 11.9 - Use a power series to approximate the definite...Ch. 11.9 - Use the result of Example 7 to compute arctan 0.2...Ch. 11.9 - Show that the function f(x)=n=0(1)nx2n(2n)! is a...Ch. 11.9 - a Show that J0 the Bessel function of order 0...Ch. 11.9 - The Bessel function of order 1 is defined by...Ch. 11.9 - a Show that the function f(x)=n=0xnn! is a...Ch. 11.9 - Let fn(x)=(sinnx)/n2. Show that the series fn(x)...Ch. 11.9 - Let f(x)=n=1xnn2 Find the intervals of convergence...Ch. 11.9 - a Starting with the geometric series n=0xn, find...Ch. 11.9 - Use the power series for tan1x to prove the...Ch. 11.9 - a By completing the square, show that...Ch. 11.10 - If f(x)=n=0bn(x5)n for all x, write a formula for...Ch. 11.10 - The graph of f is shown. a Explain why the series...Ch. 11.10 - If f(n)(0)=(n+1)! for n=0,1,2,..., find the...Ch. 11.10 - Find the Taylor series for f centered at 4 if...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Use the definition of a Taylor series to find the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Maclaurin series for f(x) using the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Find the Taylor series for f(x) centered at the...Ch. 11.10 - Prove that the series obtained in Exercise 13...Ch. 11.10 - Prove that the series obtained in Exercise 25...Ch. 11.10 - Prove that the series obtained in Exercise 17...Ch. 11.10 - Prove that the series obtained in Exercise 18...Ch. 11.10 - Use the binomial series to expand the function as...Ch. 11.10 - Use the binomial series to expand the function as...Ch. 11.10 - Use the binomial series to expand the function as...Ch. 11.10 - Use the binomial series to expand the function as...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Use a Maclaurin series in Table 1 to obtain the...Ch. 11.10 - Find the Maclaurin series of f by any method and...Ch. 11.10 - Find the Maclaurin series of f by any method and...Ch. 11.10 - Find the Maclaurin series of f by any method and...Ch. 11.10 - Find the Maclaurin series of f by any method and...Ch. 11.10 - Use the Maclaurin series for cos x to compute cos...Ch. 11.10 - Use the Maclaurin series for ex to calculate 1/e10...Ch. 11.10 - a Use the binomial series to expand 1/1x2. b Use...Ch. 11.10 - a Expand 1/1+x4 as a power series. b Use part a to...Ch. 11.10 - Evaluate the indefinite integral as an infinite...Ch. 11.10 - Evaluate the indefinite integral as an infinite...Ch. 11.10 - Evaluate the indefinite integral as an infinite...Ch. 11.10 - Evaluate the indefinite integral as an infinite...Ch. 11.10 - Use series to approximate the definite integral to...Ch. 11.10 - Use series to approximate the definite integral to...Ch. 11.10 - Use series to approximate the definite integral to...Ch. 11.10 - Use series to approximate the definite integral to...Ch. 11.10 - Use series to evaluate the limit. limx0xln(1+x)x2Ch. 11.10 - Use series to evaluate the limit. limx01cosx1+xexCh. 11.10 - Use series to evaluate the limit....Ch. 11.10 - Use series to evaluate the limit. limx01+x112xx2Ch. 11.10 - Use series to evaluate the limit....Ch. 11.10 - Use the series in Example 13b to evaluate...Ch. 11.10 - Use multiplication or division of power series to...Ch. 11.10 - Use multiplication or division of power series to...Ch. 11.10 - Use multiplication or division of power series to...Ch. 11.10 - Use multiplication or division of power series to...Ch. 11.10 - Use multiplication or division of power series to...Ch. 11.10 - Use multiplication or division of power series to...Ch. 11.10 - Find the sum of the series. n=0(1)nx4nn!Ch. 11.10 - Find the sum of the series. n=0(1)n2n62n(2n)!Ch. 11.10 - Find the sum of the series. n=0(1)n13nn5nCh. 11.10 - Find the sum of the series. n=03n5nn!Ch. 11.10 - Find the sum of the series....Ch. 11.10 - Find the sum of the series....Ch. 11.10 - Find the sum of the series. 3+92!+273!+814!+...Ch. 11.10 - Find the sum of the series. 1121323+15251727+...Ch. 11.10 - Show that if p is an nth-degree polynomial, then...Ch. 11.10 - If f(x)=(1+x3)30, what is f(58)(0)?Ch. 11.10 - Prove Taylors Inequality for n=2, that is, prove...Ch. 11.10 - a Show that the function defined by...Ch. 11.10 - Use the following steps to prove 17. a Let...Ch. 11.10 - In Exercise 10.2.53 it was shown that the length...Ch. 11.11 - a Find the Taylor polynomials up to degree 5 for...Ch. 11.11 - a Find the Taylor polynomials up to degree 3 for...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Find the Taylor polynomial T3(x) for the function...Ch. 11.11 - Use a computer algebra system to find the Taylor...Ch. 11.11 - Use a computer algebra system to find the Taylor...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - a Approximate f by a Taylor polynomial with degree...Ch. 11.11 - Use the information from Exercise 5 to estimate...Ch. 11.11 - Use the information from Exercise 16 to estimate...Ch. 11.11 - Use Taylors Inequality to determine the number of...Ch. 11.11 - How many terms of the Maclaurin series for ln(1+x)...Ch. 11.11 - Use the Alternating Series Estimation Theorem or...Ch. 11.11 - Use the Alternating Series Estimation Theorem or...Ch. 11.11 - Use the Alternating Series Estimation Theorem or...Ch. 11.11 - Suppose you know that f(n)(4)=(1)nn!3n(n+1) and...Ch. 11.11 - A car is moving with speed 20 m/s and acceleration...Ch. 11.11 - The resistivity of a conducting wire is the...Ch. 11.11 - An electric dipole consists of two electric...Ch. 11.11 - a Derive Equation 3 for Gaussian optics from...Ch. 11.11 - If a water wave with length L moves with velocity...Ch. 11.11 - A uniformly charged disk has radius R and surface...Ch. 11.11 - If a surveyor measures differences in elevation...Ch. 11.11 - The period of a pendulum with length L that makes...Ch. 11.11 - In Section 3.8 we considered Newtons method for...Ch. 11.R - a What is a convergent sequence? b What is a...Ch. 11.R - a What is a bounded sequence? b What is a...Ch. 11.R - a What is a geometric series? Under what...Ch. 11.R - Suppose an=3 and sn is the nth partial sum of the...Ch. 11.R - State the following. a The Test for Divergence b...Ch. 11.R - a What is an absolutely convergent series? b What...Ch. 11.R - a If a series is convergent by the Integral Test,...Ch. 11.R - a Write the general form of a power series. b What...Ch. 11.R - Suppose f(x) is the sum of a power series with...Ch. 11.R - a Write an expression for the nth-degree Taylor...Ch. 11.R - Write the Maclaurin series and the interval of...Ch. 11.R - Write the binomial series expansion of (1+x)k.What...Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the statement is true or false....Ch. 11.R - Determine whether the sequence is convergent or...Ch. 11.R - Determine whether the sequence is convergent or...Ch. 11.R - Determine whether the sequence is convergent or...Ch. 11.R - Determine whether the sequence is convergent or...Ch. 11.R - Determine whether the sequence is convergent or...Ch. 11.R - Determine whether the sequence is convergent or...Ch. 11.R - Determine whether the sequence is convergent or...Ch. 11.R - Determine whether the sequence is convergent or...Ch. 11.R - A sequence is defined recursively by the equations...Ch. 11.R - Show that limnn4en=0 and use a graph to find the...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is convergent or...Ch. 11.R - Determine whether the series is conditionally...Ch. 11.R - Determine whether the series is conditionally...Ch. 11.R - Determine whether the series is conditionally...Ch. 11.R - Determine whether the series is conditionally...Ch. 11.R - Find the sum of the series. n=11(3)n123nCh. 11.R - Find the sum of the series. n=11n(n+3)Ch. 11.R - Find the sum of the series. n=1[tan1(n+1)tan1n]Ch. 11.R - Find the sum of the series. n=0(1)nn32n(2n)!Ch. 11.R - Find the sum of the series. 1e+e22!e33!+e44!Ch. 11.R - Express the repeating decimal 4.17326326326. . ....Ch. 11.R - Show that coshx1+12x2 for all x.Ch. 11.R - For what values of x does the series n1(lnx)n...Ch. 11.R - Find the sum of the series n=1(1)n+1n5 correct to...Ch. 11.R - a Find the partial sum s5 of the series n=11/n6...Ch. 11.R - Use the sum of the first eight terms to...Ch. 11.R - a Show that the series n=1nn(2n)! is convergent. b...Ch. 11.R - Prove that if the series n=1an is absolutely...Ch. 11.R - Find the radius of convergence and interval of...Ch. 11.R - Find the radius of convergence and interval of...Ch. 11.R - Find the radius of convergence and interval of...Ch. 11.R - Find the radius of convergence and interval of...Ch. 11.R - Find the radius of convergence of this series...Ch. 11.R - Find the Taylor series of f(x)=sinxata=/6.Ch. 11.R - Find the Taylor series of f(x)=cosxata=/3.Ch. 11.R - Find the Maclaurin series for f and its radius of...Ch. 11.R - Find the Maclaurin series for f and its radius of...Ch. 11.R - Find the Maclaurin series for f and its radius of...Ch. 11.R - Find the Maclaurin series for f and its radius of...Ch. 11.R - Find the Maclaurin series for f and its radius of...Ch. 11.R - Find the Maclaurin series for f and its radius of...Ch. 11.R - Find the Maclaurin series for f and its radius of...Ch. 11.R - Find the Maclaurin series for f and its radius of...Ch. 11.R - Evaluate exxdx as an infinite series.Ch. 11.R - Use series to approximate 011+x4dx correct to two...Ch. 11.R - a Approximate f by a Taylor polynomial with degree...Ch. 11.R - a Approximate f by a Taylor polynomial with degree...Ch. 11.R - Use series to evaluate the following limit....Ch. 11.R - The force due to gravity on an object with mass m...Ch. 11.R - Suppose that f(x)=n=0cnxn for all x. a If f is an...Ch. 11.R - If f(x)=ex2, show that f(2n)(0)=(2n)!n!.Ch. 11.P - If f(x)=sin(x3), find f(15)(0).Ch. 11.P - A function f is defined by f(x)=limnx2n1x2n+1...Ch. 11.P - a Show that tan12x=cot12x2cotx. b Find the sum of...Ch. 11.P - Let {Pn} be a sequence of points determined as in...Ch. 11.P - To construct the snowflake curve, start with an...Ch. 11.P - Find the sum of the series...Ch. 11.P - a Show that for xy1, arctanxarctany=arctanxy1+xy...Ch. 11.P - a Prove a formula similar to the one in Problem 7a...Ch. 11.P - Use the result of Problem 7a to find the sum of...Ch. 11.P - If a0+a1+a2+...+ak=0, show that...Ch. 11.P - Find the interval of convergence of n=1n3xn and...Ch. 11.P - Suppose you have a large supply of books, all the...Ch. 11.P - Find the sum of the series n=2(11n2).Ch. 11.P - If p1, evaluate the expression...Ch. 11.P - Suppose that circles of equal diameter are packed...Ch. 11.P - A sequence {an} is defined recursively by the...Ch. 11.P - If the curve ex10sinx, x0, is rotated about the...Ch. 11.P - Starting with the vertices P1(0,1), P2(1,1),...Ch. 11.P - Find the sum of the series n=1(1)n(2n+1)3n.Ch. 11.P - Carry out the following steps to show that...Ch. 11.P - Find all the solutions of the equation...Ch. 11.P - Right-angled triangles are constructed as in the...Ch. 11.P - Consider the series whose terms are the...Ch. 11.P - a Show that the Maclaurin series of the function...Ch. 11.P - Let u=1+x33!+x66!+x99!+......Ch. 11.P - Prove that if n1, the nth partial sum of the...

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South Shore Construction builds permanent docks and seawalls along the southern shore of Long Island, New York....

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

Population Model The differential equation dpdt=(kcost)P, where k is a positive constant, is a model of human p...

A First Course in Differential Equations with Modeling Applications (MindTap Course List)

A school newspaper reporter decides to randomly survey 12 Students to see If they will attend Tet (Vietnamese N...

Introductory Statistics

For the following exercises, find the domain, range, and all zeros/intercepts, if any, of the functions. 21. g(...

Calculus Volume 1