(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.

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# Real Analysis - Radius of Convergence

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