I was wondering why this works and if it works every time

[end3r7]

Homework Statement

I have to find for which values of x the following converge

Homework Equations

$$\sum n x^{n}$$

$$\sum \frac{x^{n}}{n}$$

$$\sum n^{n} x^{x}$$

The Attempt at a Solution

I used the ratio test for the first two and the root test for the last and found respectively that x must lie within

(-1,1)
[-1,1)
{0}

Are these right?
and
Is this the right way to go all the time?

I was wondering when I'll be able to apply the ratio/root tests for radii of convergence. Is it just when we have power functions?

 

[Dick]

For the last one you mean sum(n^n*x^n), right? I think they are all correct. You can apply any legitimate test to any series, whether it is a power function or not.

 

[HallsofIvy]

The ratio and root test work for any series, as Dick said, but it may be difficult to calculate the ratios or roots. It is when you have products or powers, as in power series, that they are easy to apply.

 

[end3r7]

You are correct Dick.

And thanks for the advice guys. =D