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Math Help - check for convergence using integrals

  1. #1
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    check for convergence using integrals

    <br />
\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.
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  2. #2
    o_O
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    \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.
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