Test the following series for convergence or divergence. Justify the decision.
![\[ \sum_{n=1}^{\infty} ne^{-n^2}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-c7e2e12cc9f6f879757b7f551106bdbe_l3.png "Rendered by QuickLaTeX.com")
The series converges since
![\[ \sum_{n=1}^{\infty} ne^{-n^2} = \sum_{n=1}^{\infty} \frac{n}{e^{n^2}} \leq \sum_{n=1}^{\infty} \frac{n}{2^n}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-4de5b021857518e177e982ee3309dd28_l3.png "Rendered by QuickLaTeX.com")
We know the last series converges from a previous exercise (Section 10.14, Exercise #3).
Test the following series for convergence or divergence. Justify the decision.
The series converges since
We know the last series converges from a previous exercise (Section 10.14, Exercise #3).