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## Homework Statement

Determine whether Ʃ(1→∞) n^2/e^n converges or diverges.

## Homework Equations

L = lim (n→∞) abs [a_n+1/a_n]

## The Attempt at a Solution

The prof was out of town so left us a "self-study" task. We're looking at the Ratio Test and I want to see if my methodology is correct.

a_n+1 = (n+1)^2/e^n+1 = n^2 + 2n + 1/e^n+1

a_n = n^2/e^n

L = lim (n→∞) abs [a_n+1/a_n]

L = lim (n→∞) abs [n^2 + 2n + 1/e^(n+1) / n^2/e^n]

L = lim (n→∞) abs [n^2 + 2n + 1 / (e*n^2)]

L = lim (n→∞) abs [(n^2/n^2 + 2n / n^2 + 1/n^2)/(e*n^2/n^2)] <--- ?

L = lim (n→∞) abs [(1 + 1/n + 1/n^2) / e]

L = 1/e

The spot I'm not sure about is dividing the terms by the highest term in the denominator. When I did that for the next problem (n/2^n) I got L = .5 and it can't possibly converge to .5 because the first two terms are .5.

Does the existence of the limit simply mean convergence and not what the limit is converging to? I seem to remember something like that, but can't recall for sure.

Thanks for the help.