(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that [tex]\sum[/tex]a_{n}x^{n}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.

2. Relevant equations

3. The attempt at a solution

Since the series converges at R, then I know that [tex]\sum[/tex]a_{n}R^{n}= M.

At -R, the series is the following: [tex]\sum[/tex]a_{n}(-R)^{n}= [tex]\sum[/tex](-1)^{n}a_{n}R^{n}.

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Real Analysis - Radius of Convergence

**Physics Forums | Science Articles, Homework Help, Discussion**