1. Infinite series

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

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

2. 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.

3. Originally Posted by Em Yeu Anh
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