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

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

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- Feb 6th 2010, 11:29 AMEm Yeu AnhInfinite series
Determine if this series is convergent or divergent. If it converges, find the sum of the series.

$\displaystyle \sum_{n=1}^{\infty}(\frac{1}{e^n}+\frac{1}{n(n+2)} ) $ - Feb 6th 2010, 12:12 PMfelper
You can separate the serie in two other series. The first, is a geometric serie $\displaystyle (e^{-1})^n$. The second converges by p-series test.

- Feb 6th 2010, 12:22 PMskeeter
note that this sum can be written as the sum of two series ...

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

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 $\displaystyle \frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{n} - \frac{1}{n+2}$ , a convergent telescoping series.

Telescoping series - Wikipedia, the free encyclopedia