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Math Help - converges

  1. #1
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    converges

    I need to find the value of this or show that it does not converge:
    The summation from n=1 to infinity of <br />
\frac{{2^{n+1}+3^{n-1}}}{{5^n}}<br />
    Any help would be very appreciated.
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  2. #2
    Eater of Worlds
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    Rewrite as:

    2\sum_{n=1}^{\infty}(\frac{2}{5})^{n}+\frac{1}{3}\  sum_{n=1}^{\infty}(\frac{3}{5})^{n}

    Now, the sums are easier. It's just a matter of a geometric series.

    For instance, \sum_{n=1}^{\infty}(\frac{2}{5})^{n}=\frac{\frac{2  }{5}}{1-\frac{2}{5}}=\frac{2}{3}

    Can you finish now?.
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  3. #3
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    Okay, that helps a lot.
    So after splitting it up like you did <br />
2\sum_{n=1}^{\infty}(\frac{2}{5})^{n}+\frac{1}{3}\  sum_{n=1}^{\infty}(\frac{3}{5})^{n}<br />
    I can rewrite this as \frac{4}{3}+\frac{1}{2}=\frac{11}{6}
    Is that correct?
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  4. #4
    Eater of Worlds
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    Yep. That's right.
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