What is the Interval of Convergence for These Power Series?

[fsm]

I just wanted to see if someone could verify my answers:

$$\\sum_{n=0}^\\infty \\frac{(x+7)^n}{sqrt(n)}$$
I get:
-8<x<-6

$$\\sum_{n=1}^\\infty \\frac{(-1)^n*x^2n}{n!}$$
This one I'm not sure of. When I take the limit I get 0. When I solve the inequality I get x. I can't find an example of this.

 

[fsm]

This might be better:

 

[courtrigrad]

1. correct
2. $$\sum_{n=1}^{\infty} \frac{(-1)^{n}x^{2n}}{n!}$$$$|\frac{(-1)^{n+1}x^{2n+1}}{n!(n+1)}\frac{n!}{(-1)^{n}x^{2n}} = \frac{x}{n+1} \rightarrow 0$$. Thus the interval of convergence is $$(-\infty, \infty)$$

 

[fsm]

I don't understand your answer for #2. I got the same thing but how is this the interval of convergence?

 

[d_leet]

I don't understand your answer for #2. I got the same thing but how is this the interval of convergence?

What don't you understand about it?

 

[fsm]

All the stuff to the right of the equal sign now just appeared. Thanks for the help.