2. Consider the seriesn=1n+1an where =an n²(n + 2)²*k(a) Find a formula for the kth partial sum, sk=an. (Hint: partial fractions!)n=1(b) Hence show that the series converges, and find its limit.

Answered: Image /qna-images/answer/c0125aba-3359-4932-8d07-01e8a9ea682c.jpg

Answered: 2. Consider the series [ª 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
4. Consider the series n=1 an where an = 1 n(n+3)* (a) Find a formula for the kth partial sum, sk = k an. (Hint: partial fractions!) n=1 (b) Hence show that the series converges, and find its limit.
6. Let a be a real number. Consider the series where an = cos(na): 2n +1 n=0 (a) Is it possible to find an a > 0 such that the above series is both absolutely convergent and conditionally convergent? Briefly explain your reasoning. (b) Find all a > 0 such that the series diverges. (c) Find all a > 0 such that the series converges absolutely. (d) Find all a > 0 such that the series converges conditionally.
11. Practice similar Help me with this By recognizing the series sum = −((3/4)) – as a Taylor series evaluated at a particular value of x, find the sum of the convergent series. ((3/4))² ((3/4))³ 2 ▶ ((3/4))" n
2. Consider the power series > ak(x + 2)*. Assume that we know that the series ak converges, but that k=0 k=0 ar(4*) diverges. k=0 (a) Does the power series converge at x = -1.3? Explain how you know, or explain why you don't have enough information to answer the question. (b) Does the series > ak (5)* converge or diverge? Explain how you know, or explain why you don't have k=0 enough information to answer the question.
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!
[41] Consider the power series in number 3 and use the previous item to find its interval of convergence.

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2. Consider the series [ª an where an = n+1 n²(n + 2)²* k (a) Find a formula for the kth partial sum, sk=an. (Hint: partial fractions!) n=1 (b) Hence show that the series converges, and find its limit.