Finding the Sum of a Convergent Series with Partial Fractions

[jwxie]

Homework Statement

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 )

Homework Equations

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?

 

[Mark44]

Homework Statement

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 )

Homework Equations

The Attempt at a Solution

Well, the kth test for divergence said this series has limit ak = 0

So the k-th term test for divergence tells us nothing.

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?

 

[jwxie]

wait. kth term test said if lim ak =/= 0, it must diverges. if it is zero, it may or may not diverge.

So i just that to begin with. Now I need a different approach to find the sum. It converges to 1. I mean I can approximate it goes goes to 1 if I look at the pattern numbers. But definitely I can't be sure.

How would I compute it?

 

[Mark44]

Homework Statement

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 )

Homework Equations

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.

Yes, lim a_k = 0, which means that the k-th term test does not apply. You can't just simplify the leading terms, and this series is not 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,

?
Carry this out. Find A, B, C, and D.

so after mulitiplication, we have
2k + 1 = Ak + 2Bk + B

No, this is not how it works. On the right side you are going to have terms up to degree 3. The coefficients of the k³ and k² terms have to be zero, but you can't just ignore them.

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?

If you complete your partial fractions work, you will find that this is a telescoping series.

wait. kth term test said if lim ak =/= 0, it must diverges. if it is zero, it may or may not diverge.

And this tells us exactly nothing.

So i just that to begin with. Now I need a different approach to find the sum. It converges to 1. I mean I can approximate it goes goes to 1 if I look at the pattern numbers. But definitely I can't be sure.

How would I compute it?