# Thread: Match each of the following with the correct statement.

1. ## Match each of the following with the correct statement.

Match each of the following with the correct statement.
C stands for Convergent, D stands for Divergent.

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2. Originally Posted by ewkimchi
Match each of the following with the correct statement.
C stands for Convergent, D stands for Divergent.

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1. $\displaystyle \frac{8 + 9^n}{9^n} = \frac{8}{9^n} + 1$

So $\displaystyle \lim_{n \to \infty}\frac{8 + 9^n}{9^n} = \lim_{n \to \infty}\left(\frac{8}{9^n} + 1\right)$

$\displaystyle = 1$

$\displaystyle > 0$.

So the series is divergent.

3. Originally Posted by ewkimchi
Match each of the following with the correct statement.
C stands for Convergent, D stands for Divergent.

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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.