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I have the following statement:

If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R_{1}$ and $\sum_{n=0}^{\infty}b_{n}x^n $ converges for $|x| < R_{2}$ , then $\sum_{n=0}^{\infty}(a_{n}+b_{n})x^n $ converges for $|x| < R_{1}+R_{2}$ as well.

I have tried to disprove this statement for a long time but couldnt get to a final answer. Thanks alot!

GeorgeB
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  • Related http://math.stackexchange.com/questions/309466/radius-of-convergence-of-a-sum-of-power-series –  Jun 01 '16 at 10:54
  • To find a counterexample, try letting both series be the same (say, $a_n=b_n=1$ for all $n$ for example). – Matthew Towers Jun 01 '16 at 10:56

1 Answers1

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Take $ a_n = 0 $ for every $ n $ then $ R_1 = +\infty $ and $ b_n $ such that $ R_2 <\infty $ (ex. $ b_n = 1 $ so $ R_2 = 1$). So $ R_1 +R_2 = +\infty $ which is a contradiction to the fact that

$ \sum (a_n+b_n)x^n = \sum b_n x^n $ so $ R_1 + R_2 = R_2 = +\infty.$

user318989
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