4. Consider the seriesn=1an wherean =1n(n+3)*(a) Find a formula for the kth partial sum, sk =kan. (Hint: partial fractions!)n=1(b) Hence show that the series converges, and find its limit.

Answered: Image /qna-images/answer/c71452e7-9cc4-432c-a520-bf9eefea445a.jpg

Answered: 4. Consider the series an where an =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the series 1/3 +1/9+1+27+1+81 +...... the first partial sums are s1=1/3, s2 =4/9, s3=13/27, s4=40/81. Determine the numerator of the nth partial sum sn= ???/3n. What can you say about the series? Explain
Consider the series San where n=1 (-1)"n n² + 2n + 5 In this problem you must attempt to use the Ratio Test to decide whether the series converges. Compute an = an+1 L = lim n-00 an Enter the numerical value of the limit L if it converges, INF if it diverges to infinity, -INF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity. L = Which of the following statements is true? A. The Ratio Test says that the series converges absolutely. B. The Ratio Test says that the series diverges. C. The Ratio Test says that the series converges conditionally. D. The Ratio Test is inconclusive, but the series converges absolutely by another test or tests. E. The Ratio Test is inconclusive, but the series diverges by another test or tests. F. The Ratio Test is inconclusive, but the series converges conditionally by another test or tests. Enter the letter for your choice here:
Consider the series of image. (a) Compute the first eight terms of the series of partial sums. It's possiblewhether this series is convergent or divergent? Justify your answer.(b) If convergent, use the Alternate Series Estimation Theorem toestimate the correct sum to four decimal places.
Calculate the sum of the series, if divergent please specify.
5. Consider the series n! n=0 for real numbers x. (a) Use the ratio test to determine the values x for which the series is absolutely convergent. (b) Does this mean that the series is convergent for these values of x? Explain. (c) Explain how one can use part (b) above to determine the limit of the sequence {an} where 22n an n!
a) Determine if the geometric series below is convergent or not. 00 2" -1 Σ n =1 5h +1 The series is (Insert C if you think it is convergent and D if you think it is divergent) b) Find the sum of the series 8" -1 Σ n =1 9n +1 The sum = c) Find the partial sum S3 of the series 8-1 Σ n=1 9h +1 The partial sum S3=

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4. Consider the series an where an = n=1 1 n(n+3) k (a) Find a formula for the kth partial sum, sk. = n=1 (b) Hence show that the series converges, and find its limit. an. (Hint: partial fractions!)