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.