Real Analysis - Radius of Convergence

[steelphantom]

Homework Statement

Suppose that \suma_nxⁿ has finite radius of convergence R and that a_n >= 0 for all n. Show that if the series converges at R, then it also converges at -R.

Homework Equations

The Attempt at a Solution

Since the series converges at R, then I know that \suma_nRⁿ = M.

At -R, the series is the following: \suma_n(-R)ⁿ = \sum(-1)ⁿa_nRⁿ.

I'm not sure where to go from here. I thought I needed to use the alternating series test, but how can I know that a1 >= a2 >= ... >= an for all n? Do I know this because the series converges? Thanks for your help.

 

[Dick]

Right. You can't use the alternating series test. How about a comparison test?

 

[steelphantom]

Thanks! So since \suma_nRⁿ converges, and a_n(-R)ⁿ <= a_nRⁿ for all n, then \suma_n(-R)ⁿ converges.

 

[Dick]

Sorry! That's wrong. I'm clearly asleep at the wheel. That's convergence for sequences. And this sort of argument only shows that the partial sums are bounded, not that they converge. Do you know the Dirichlet convergence test?

 

[steelphantom]

Sorry! That's wrong. I'm clearly asleep at the wheel. That's convergence for sequences. And this sort of argument only shows that the partial sums are bounded, not that they converge. Do you know the Dirichlet convergence test?

I don't know that one. But the comparison test in my book says the following:

Let \suma_n be a series where a_n >=0 for all n.
(i) If \suma_n converges and |b_n| <= a_n for all n, then \sumb_n converges.

If I let a_n = a_nRⁿ, this is >=0 for all n. And if I let b_n = a_n(-R)ⁿ, then I have |b_n| <= a_n for all n, so the series converges, right? What's wrong with this statement?

 

[Dick]

Nothing wrong with that. Unfortunately, I wasn't thinking of that comparison test. Hence the panic attack. Carry on.

 

[steelphantom]

Ok! Thanks once again for your help.

 

[HallsofIvy]

What's wrong with using the comparison test is that it only applies to positive series. Certainly -n< (1/2)ⁿ for all n, but we can't use that to conclude that \sum -n converges!

The crucial point is that every a_n is positive. That means that \sum a_n x^n, for x negative is an alternating series. What is true of alternating series?

 

[steelphantom]

The crucial point is that every a_n is positive. That means that \sum a_n x^n, for x negative is an alternating series. What is true of alternating series?

If a1 >= a2 >= ... >= an >= ... >= 0 and lim(an) = 0, then the alternating series \sum(-1)ⁿa_n converges. But, like I said before, do I know that a1 >= a2 >= ... >= an because the series converges at R?

 

[Dick]

What's wrong with using the comparison test is that it only applies to positive series. Certainly -n< (1/2)ⁿ for all n, but we can't use that to conclude that \sum -n converges!

The crucial point is that every a_n is positive. That means that \sum a_n x^n, for x negative is an alternating series. What is true of alternating series?

All of terms a_n*R^n are positive and it's convergent. The series is absolutely convergent. Nothing can go wrong here.