Question

find if the series is convergent and what it sums to

the sum from k=3 to infinity of (k+1)^2/((k-1)(K-2))

I'm not sure how to start

Answer 1

How does (k+1)^2/((k-1)(K-2)) behave for large k?

Answer 2

isn't it infinity/infinity, which is undefined?

Answer 3

It's infinity/infinity but you can compute the limit, which is finite. Som try to compiute the limit and show that it is nonzero. Then you say that because the limit is not zero the series cannot converge.

Note that even if the limit were zero, that would still not guarantee convergence of the series. The terms have to approach zero fast enough...

Answer 4

how would you compute the limit?
when i do it on my calculator i find the limit is undefined...what does that say for the sum?

Answer 5

(k+1)^2/((k-1)(K-2)) =

[k^2 + 2 k + 1]/[k^2 - 3 k + 2] =

[1 + 2/k + 1/k^2]/[1 - 3/k + 2/k^2]

Can you see what happens if k --> infinity?

Answer 6

is it approaching 0?

Answer 7

[1 + 2/k + 1/k^2]/[1 - 3/k + 2/k^2]

Numerator is

A(k) = 1 + 2/k + 1/k^2

Denominator is:

B(k) = 1 - 3/k + 2/k^2

What are the limits of A(k) and B(k) for k ---> infinity?