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)} ) $
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