Power Series: Interval Of Convergence

[tak13]

Homework Statement

I am not really good with Series so I having a hard time with these problems.

http://img835.imageshack.us/img835/858/img1257d.jpg

Homework Equations

The Attempt at a Solution

The part I am stuck is where I highlighted. The first question: The whole thing is squared so I don't know how to do it with the squared in the way.

The second question: It is just plain weird. I think I am not supposed to do that for the highlighted part.

The answer for first question is : 4 (-4,4)
The answer for second question is : infinity (-infinity,+infinity)

 

[Square]

First:

$$\lim_{n\to \infty} \left| \frac{\left(n \cdot (n-1) \cdot (n-2) \ldots \right) ^{2} \cdot (2n+2)(2n+1)}{\left( (n+1) \cdot n \cdot (n-1) \cdot (n-2) \ldots \right) ^{2}} \right| = \lim_{n\to \infty} \left| \frac{2(n+1)(2n+1)}{(n+1)^{2}} \right| = \lim_{n\to \infty}\left| \frac{2(2n+1)}{(n+1)} \right| = 4$$

On the second one you're basically done.

$$\lim_{n\to \infty} \left| \frac{(2n+2)(2n+1)}{(2n+3)} \right| = \infty \ .$$

Think the answers are mixed up.

 

[tak13]

Haha, I just looked over the manual, the second question does seem to be like that. I guess I messed up looking at the manual the first time.

For the first one do you mean I should do like this:

http://img694.imageshack.us/img694/4568/img1259u.jpg

 

[Square]

Haha, I just looked over the manual, the second question does seem to be like that. I guess I messed up looking at the manual the first time.

For the first one do you mean I should do like this:

Yep. Factor the numerator and you'll get

$$\lim_{n\to \infty} \left| \frac{2(n+1)(2n+1)}{(n+1)^2} \right| = \lim_{n\to \infty} \left| \frac{2(2n+1)}{n+1} \right|$$

which evaluates to 4 (divide numerator and denominator with $$n$$).