Convergence of a Taylor Series: Finding the Values of x

[ThatOneGuy45]

Homework Statement

For this problem I am to find the values of x in which the series converges. I know how to do that part of testing of convergence but constructing the summation part is what I am unsure about.

I am given the follwing:
1 + 2x + [itex]\frac{3^2x^2}{2!}[/itex] +[itex]\frac{4^3x^3}{3!}[/itex]+ ...

Homework Equations

I looked up online about the taylor series expansion for e^x because I noticed it looked familiar and compared it with the series

The taylor series for e^x is:
1 + x + [itex]\frac{x^2}{2!}[/itex] +[itex]\frac{x^3}{3!}[/itex]+ ...=Ʃ[itex]^{∞}_{n=0}[/itex][itex]\frac{x^n}{n!}[/itex]

The Attempt at a Solution

What I did was pretty much just put [itex]\frac{(n+1)^nx^n}{n!}[/itex] and checked the terms to see if it works. It seems to work but I am just a bit unsure. I haven't worked with this stuff in a while so I just want to be sure if I did that part right.

 

[clamtrox]

That looks correct. You should be able to see if it converges using the ratio test, just be careful with the limit.