Testing a series for convergence/divergence

[tatiana_eggs]

Homework Statement

Use special comparison test to find if $$\frac{2+(-1)^n}{n^2+7}$$ is convergent or divergent.

Homework Equations

Special comparison test using the convergent series $$\frac{1}{n^2}$$

and taking the limit as n-> infinity of my initial series $$\frac{2+(-1)^n}{n^2+7}$$ divided by my comparison series $$\frac{1}{n^2}$$

which comes out to be lim n--> infinity of $$\frac{2n^2+(-1)^n(n^2)}{n^2+7}$$.

The Attempt at a Solution

I guess I need help evaluating that limit because what I'm getting is undefined (alternating) and the back of the book says that it's defined.

 

[Raskolnikov]

$$0 \leq \frac{2+(-1)^n}{n^2+7} \leq \frac{3}{n^2+7}.$$ You can squeeze it as such. If the series on the right converges, then so does yours.