Originally Posted by

**Paperwings** Problem

Examine the following series for convergence

1. $\displaystyle \sum_{k=1}^{\infty}ke^{-k^2}$

2. $\displaystyle \sum_{k=1}^{\infty} \left(\frac{k+1}{k^2+1}\right)^3$

=================================

Attempt:

1. This series$\displaystyle \sum_{k=1}^{\infty}ke^{-k^2}$ can be rewritten as $\displaystyle \sum_{k=1}^{\infty} \frac{k}{e^{k^2}}$.

I try to use the Ratio Test. By the Ratio Test

$\displaystyle \lim_{k \to \infty} \left| \frac{k+1}{e^{(k+1)^2}} \frac{e^{k^2}}{k}\right| = \lim_{k \to \infty} \left| \frac{e^{k^2}}{e^{(k+1)^2}} \left( 1 + \frac{1}{k}\right) \right|$

I know the term 1 + 1/k goes to 1 as $\displaystyle k \rightarrow \infty $. However I do know about the left term. $\displaystyle \frac{e^{k^2}}{e^{(k+1)^2}}$

2. I tried to find if this series converges or not by the Comparison Test but went nowhere.

Thank you for your time.