What is the radius of convergence of

[Shackleford]

Homework Statement

z ∈ ℂ

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

Homework Equations

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

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

lim_n→∞ |a_n|^1/n

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/lim_n→∞ |a_n|^1/n

 

[Dick]

Homework Statement

z ∈ ℂ

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

Homework Equations

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

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

lim_n→∞ |a_n|^1/n

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/lim_n→∞ |a_n|^1/n

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.

 

[Shackleford]

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.

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?

 

[Dick]

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?

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).