- #1

Potatochip911

- 318

- 3

## Homework Statement

Using the integral test determine whether or not the following sum diverges. $$ \sum_{n=1}^{\infty} \frac{e^n}{e^{2n}+9} $$

## Homework Equations

## The Attempt at a Solution

$$ \sum_{n=1}^{\infty} \frac{e^n}{e^{2n}+9}=\sum_{n=1}^{\infty} \frac{e^n}{({e^{n})}^2+9} \\

\int^{\infty} \frac{e^n}{({e^{n})}^2+9} \cdot dn; \hspace{0.5cm} Let \space u=e^n; \hspace{0.5cm} du=e^n\cdot dn \\

\int^{\infty} \frac{du}{u^2+9}; \hspace{0.5cm} Let \space u=3\tan\theta \hspace{0.5cm} du=3\sec^2\theta \\

\int^{(\frac{\pi}{2})^-} \frac{3\sec^2\theta}{9\tan^2\theta+9}=\frac{1}{3} \int^{(\frac{\pi}{2})^-} 1\cdot d\theta=\frac{\pi}{6} $$ Since I didn't get infinity in my answer this sum converges however my answer key says that it does diverge, did I miss something?