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All Textbook Solutions for Single Variable Calculus: Early Transcendentals
19E20E21E22EDetermine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 23. n=1(3)n14n24E25EDetermine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 26. n=1622n13n27E28E29E30E31EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. 32. n=1[(0.2)n+(0.6)n1]Determine whether the series is convergent or divergent. If it is convergent, find its sum. 33. n=114+en34E35E36EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. 37. n=1ln(n2+12n2+1)38E39E40E41EDetermine whether the series is convergent or divergent. If it is convergent, find its sum. 42. n=1enn2Determine whether the series is convergent or divergent by expressing sn as a telescoping sum (as in Example 8). If it is convergent, find its sum. 43. n=22n2144E45E46E47EDetermine whether the series is convergent or divergent by expressing sn as a telescoping sum (as in Example 8). If it is convergent, find its sum. 48. n=21n3n49E50E51EExpress the number as a ratio of integers. 52. 0.46=0.4646464653E54E55E56E57EFind the values of x for which the series converges. Find the sum of the series for those values of x. 58. n=1(x+2)n59E60E61E62E63E64E65E66E67E68E69E70EA patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, 5% of the drug remains in the body. (a) What quantity of the drug is in the body after the third tablet? After the nth tablet? (b) What quantity of the drug remains in the body in the long run?72E73E74E75E76E77E78EThe figure shows two circles C and D of radius 1 that touch at P. The line T is a common tangent line; C1 is the circle that touches C, D, and T; C2 is the circle that touches C, D, and C1; C3 is the circle that touches C, D, and C2. This procedure can be continued indefinitely and produces an infinite sequence of circles {Cn}. Find an expression for the diameter of Cn and thus provide another geometric demonstration of Example 8.80E81E82E83E84EIf an is convergent and bn is divergent, show that the series (an +bn) is divergent. [Hint: Argue by contradiction.]86E87EThe Fibonacci sequence was defined in Section 11.1 by the equations f1=1,f2=1,fn=fn1+fn2n3 Show that each of the following statements is true. (a) 1fn1fn+1=1fn1fn1fnfn+1 (b) n=21fn1fn+1=1 (c) n=2fnfn1fn+1=289E90EConsider the series n=1n/(n+1)!. (a) Find the partial sums s1, s2, s3, and s4. Do you recognize the denominators? Use the pattern to guess a formula for sn. (b) Use mathematical induction to prove your guess. (c) Show that the given infinite series is convergent, and find its sum.In the figure at the right there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.1E2E3E4EUse the Integral Test to determine whether the series is convergent or divergent. 5.n=125n16EUse the Integral Test to determine whether the series is convergent or divergent. 7. n=1nn2+18E9E10E11E12E13E14EDetermine whether the series is convergent or divergent. 15. n=1n+4n216E17E18E19EDetermine whether the series is convergent or divergent. 20. n=33n4n22n21E22E23E24E25E26E27E28E29E30E31E32EThe Riemann zeta-function is defined by (x)=n=11nx and is used in number theory to study the distribution of prime numbers. What is the domain of ?34E35E36E37E38E39E40E41E42E(a) Use (4) to show that if sn is the nth partial sum of the harmonic series, then sn1+lnn (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 22.44E45EFind all values of c for which the following series converges. n=1(cn1n+1)Suppose an and bn are series with positive terms and bn is known to be convergent. (a) If an bn for all n, what can you say about an? Why? (b) If an bn for all n, what can you say about an? Why?Suppose an and bn are series with positive terms and bn is known to be divergent. (a) If an bn for all n, what can you say about an? Why? (b) If an bn for all n, what can you say about an? Why?3E4EDetermine whether the series converges or diverges. 5. n=1n+1nnDetermine whether the series converges or diverges. 6. n=1n1n3+17E8E9E10E11E12EDetermine whether the series converges or diverges. 13. n=11+cosnenDetermine whether the series converges or diverges. 14. n=113n4+13Determine whether the series converges or diverges. 15. n=14n+13n2Determine whether the series converges or diverges. 16. n=11nnDetermine whether the series converges or diverges. 17. n=11n2+118E19EDetermine whether the series converges or diverges. 20. n=1n2+n+1n4+n221E22E23E24E25E26E27E28E29EDetermine whether the series converges or diverges. 30. n=1n!nn31E32E33E34EUse the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 35. n=15ncos2n36EThe meaning of the decimal representation of a number 0.d1d2d3 (where the digit di is one of the numbers 0, 1, 2, , 9) is that 0.d1d2d3d4=d110+d2102+d3103+d4104+ Show that this series always converges.38E39E40E41E42E43E44EIf an is a convergent series with positive terms, is it true that (sin)an is also convergent?46E(a) What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can yon say about the remainder after n terms?Test the series for convergence or divergence. 2. 2325+2729+211Test the series for convergence or divergence. 3. 25+4667+88109+4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20EGraph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places. 21. n=1(0.8)nn!22E23E24E25E26E27E28E29E30E31E32E33E34E35E36EWhat can you say about the series an in each of the fallowing cases? (a) limn|an+1an|=8 (b) limn|an+1an|=0.8 (c) limn|an+1an|=12E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18EUse the Ratio Test to determine whether the series is convergent or divergent. 19. n=1n100100nn!20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35EUse any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. 36. n=1sin(n/6)1+nnUse any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. 37. n=1(1)narctannn238EThe terms of a series are defined recursively by the equations a1=2an+I=5n+14n+3an Determine whether an converges or diverges.40E41E42E43EFor which positive integers k is the following series convergent? n=1(n!)2(kn)!(a) Show that n0xn/n! converges for all x. (b) Deduce that limn xn/n! = 0 for all x.46E47E48E49E50E51E52E53ETest the series for convergence or divergence. 1. n=1n21n3+12E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38EWhat is a power series?(a) What is the radius of convergence of a power series? How do you find it? (b) What is the interval of convergence of a power series? How do you find it?3EFind the radius of convergence and interval of convergence of the series. 4. n=1(1)nxnn35E6E7E