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Math Help - Infinite series

  1. #1
    Member Em Yeu Anh's Avatar
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    Red face Infinite series

    Determine if this series is convergent or divergent. If it converges, find the sum of the series.

    \sum_{n=1}^{\infty}(\frac{1}{e^n}+\frac{1}{n(n+2)}  )
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  2. #2
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    You can separate the serie in two other series. The first, is a geometric serie (e^{-1})^n. The second converges by p-series test.
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  3. #3
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    Quote Originally Posted by Em Yeu Anh View Post
    Determine if this series is convergent or divergent. If it converges, find the sum of the series.

    \sum_{n=1}^{\infty}(\frac{1}{e^n}+\frac{1}{n(n+2)}  )
    note that this sum can be written as the sum of two series ...

    \sum_{n=1}^{\infty} \frac{1}{e^n}+\frac{1}{n(n+2)} = \sum_{n=1}^{\infty} \left(\frac{1}{e}\right)^n + \sum_{n=1}^{\infty} \frac{1}{n(n+2)}

    the first sum is a convergent geometric series.

    Geometric series - Wikipedia, the free encyclopedia


    using the method of partial fractions, the second can be written as \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n} - \frac{1}{n+2} , a convergent telescoping series.

    Telescoping series - Wikipedia, the free encyclopedia
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