# Prove that a series converges.

1. Dec 1, 2012

### anoncow

1. The problem statement, all variables and given/known data
$$\sum\limits_{n=1}^\infty \frac{1 - (-1)^n e}{1 + (n \pi)^2}$$
2. Relevant equations

3. The attempt at a solution
I'd imagine you have to use the comparison test on it but am unable to figure out where to start. Any suggestions would be greatly appreciated.

Last edited: Dec 1, 2012
2. Dec 1, 2012

### MathematicalPhysicist

Well indeed the comparison test will do (with 1/n^2).

I'll let you see how exactly.

3. Dec 1, 2012

### lurflurf

Interesting sum, compare to 1/n^2

4. Dec 1, 2012

### anoncow

Last edited: Dec 1, 2012
5. Dec 1, 2012

### MathematicalPhysicist

Much simpler than that:

$$|\frac{1-(-1)^ne}{1+(\pi n)^2}| \leq \frac{1+e}{1+(\pi n)^2}$$

6. Dec 1, 2012

### anoncow

oh so could i just have done:

$$0 \leq |\frac{1-(-1)^ne}{1+(\pi n)^2}| \leq |\frac{1}{n^2}|$$
$$\therefore \frac{1-(-1)^ne}{1+(\pi n)^2}$$ absolutely converges.