(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Determine whether the series converges or diverges. For convergent series, find the sum of the series.

sima (k=1, infinity), (2k +1) / ((k^2) (k+1)^2 )

2. Relevant equations

3. The attempt at a solution

Well, the kth test for divergence said this series has limit ak = 0 because if we simplify the leading terms, we have 2k / c*k^4, which is 1/k^3, this is a p-series. We know that for p > 1, the p-series will converge.

But how do you find the sum?

Partial fraction expanision seems not a good choice?

(2k +1) / ((k^2) (k+1)^2 ) = A/k + B/k^2 + C/(k+1) + D/(k+1)^2

->>>>

(2k+1) = A(k)(k+1)^2 + B(k+1)^2 + C(k^2)(k+1) + D(k^2)

We can ignore anything that has power higher than 1, so after mulitiplication, we have

2k + 1 = Ak + 2Bk + B

it's clear that b = 1

2k = k(a+2b)

2k = k(a+2)

2 = a + 2

a = 0

so are we left with the expansion 1/k^2 ? which doesn't make sense to me...

well anyhow, even if i have a compute error, this is not a telescopic series, no positive terms will cancel. How do I compute the sum then?

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# Homework Help: Find the sum of this series

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