Prove that a series converges.

[anoncow]

Homework Statement

$$ \sum\limits_{n=1}^\infty \frac{1 - (-1)^n e}{1 + (n \pi)^2}$$

Homework Equations

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.

 

[MathematicalPhysicist]

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

I'll let you see how exactly.

 

[lurflurf]

Interesting sum, compare to 1/n^2

 

[anoncow]

Is this sufficient proof?

http://mathb.in/1308?key=f6d1a50cb6e98de9513ed7740481e9387911e3ce

 

[MathematicalPhysicist]

Much simpler than that:

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

 

[anoncow]

Much simpler than that:

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

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.