# Math Help - check for convergence using integrals

1. ## check for convergence using integrals

$
\sum\frac{e^n}{1+e^{2n}}$
. i don't know how to add the number of the sum, but it's 1 to $\infty$. it's a decreasing series, but i can't find the integral.
it looks like it may turn into $\frac{\infty}{\infty}$ using l'hopital's rule and i just don't know what to do with it.

2. $\int \frac{e^x}{1 + e^{2x}} \ dx = \int \frac{e^x}{1 + \left(e^x\right)^2} \ dx$

Let: ${\color{red}u = e^x} \ \Rightarrow \ {\color{blue} du = e^x \ dx}$

So: $\int \frac{{\color{blue}e^x}}{{\color{red}1 + \left(e^x\right)^2}} \ {\color{blue}dx} = \int \frac{{\color{blue}du}}{1 + {\color{red}u}^2}$

which is a standard integral.