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I have a question.

I want to know an example of a power series centered at $x = 0$ that converges on $[-2,2]$ but not absolutely on the entire interval $[-2,2]$, and diverges otherwise.

I saw that it could be $\frac{(-1)^k}{k}x^{k(k+1)}$. But I don't know how you can say that it converge on $[-2,2]$ because with the ratio test or the root test I get only an interval of $(-1,1)$? And do they mean that it is not absolutely convergent on the entire interval $[-2,2]$ that it converge conditionally?

Can someone help me? Thank you

joesingo
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C...
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  • Also, you need to be careful with the ratio test when your series is missing terms. It doesn't work how you would expect. – Hans Musgrave Feb 04 '17 at 14:25
  • @lulu Can I say that I fill instead of x (x/2) in your formula? And is the formula I was given correct? – C... Feb 04 '17 at 14:28
  • Yes, you can just take $\frac x2$ in my formula, $\sum (-1)^n \frac {x^{2n}}n$. Or in your formula, for that matter. – lulu Feb 04 '17 at 14:47

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If you want convergence on $[-2,2]$ then you should scale $x$ by a factor of two in your example. In order to get conditional convergence at $\{-2,2\}$ you may e.g. take: $$ f(x) = \ln(1+(x/2)^2) = \sum_{k\geq 1} \frac{(-1)^{k-1}}{4^k k} x^{2k} $$ (the point being that I have put the singularity into the complex plane).

H. H. Rugh
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