# Match each of the following with the correct statement.

#### ewkimchi

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

1.

2.

3.

4.

5.

#### Prove It

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

1.

2.

3.

4.

5.
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.

HallsofIvy

#### Prove It

MHF Helper
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.

HallsofIvy