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CalculusCalculus (MindTap Course List)(a) Derive Equation 3 for Gaussian optics from Equation 1 by approximating cos ϕ in Equation 2 by its first-degree Taylor polynomial. (b) Show that if cos ϕ is replaced by its third-degree Taylor polynomial in Equation 2, then Equation 1 becomes Equation 4 for third-order optics. [ Hint: Use the first two terms in the binomial series for l o − 1 and l i − 1 . Also, use ϕ ≈ sin ϕ ]BuyFind*arrow_forward*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 11.11, Problem 34E

Textbook Problem

(a) Derive Equation 3 for Gaussian optics from Equation 1 by approximating

(b) Show that if
*Hint:* Use the first two terms in the binomial series for

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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