# What is the radius of convergence of

1. Feb 9, 2015

### Shackleford

1. The problem statement, all variables and given/known data

z ∈ ℂ

What is the radius of convergence of (n=0 to ∞) Σ anzn?

2. Relevant equations

I used the Cauchy-Hardamard Theorem and found the lim sup of the convergent subsequences.

$$a_n = \frac{n+(-1)^n}{n^2}$$

limn→∞ |an|1/n

3. The attempt at a solution

I think that the radius of convergence is one, i.e. |z| < 1. I figured that the numerator would tend to n with the oscillating 1 and so you'd get $$\frac{n^{1/n}}{n^{2/n}} = 1$$

R = 1/limn→∞ |an|1/n

2. Feb 9, 2015

### Dick

That's a little informal but it looks fine. You might want worry about what happens at z=1 and z=(-1) if you are concerned about the boundary cases.

3. Feb 10, 2015

### Shackleford

I assume a singularity is there and outside of the disk the series is divergent. Well, I'm leaving out a few bits of information in my "proof" here. I'll state a bit more of the background. What would a more formal proof look like?

4. Feb 10, 2015

### Dick

Rather than just ignoring the (-1)^n handle it with a squeeze type thing. E.g. n/2<=n+(-1)^n<=2n. You can easily find the outer limits. Discuss what happens when z=1 or z=(-1).