Match each of the following with the correct statement.
C stands for Convergent, D stands for Divergent.
1.
2.
3.
4.
5.
4. $\displaystyle \sum_{n = 1}^{\infty}n\,e^{-n^2} \leq \int_1^{\infty}x\,e^{-x^2}\,dx$
$\displaystyle = -\frac{1}{2}\int_1^{\infty}{-2x\,e^{-x^2}\,dx}$
$\displaystyle = \lim_{\varepsilon \to \infty}-\frac{1}{2}\left[e^{-x^2}\right]_1^{\varepsilon}$
$\displaystyle =\lim_{\varepsilon \to \infty}-\frac{1}{2}\left[e^{-\varepsilon^2} - e^{-1}\right]$
$\displaystyle = -\frac{1}{2}\left[0 - e^{-1}\right]$
$\displaystyle = \frac{1}{2}\,e^{-1}$.
By the integral test, the series converges.