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Suppose that I know the series for $(\tan^{-1})'=\sum_{n=0}^{\infty} (-1)^nx^{2n}$ converges when $\lvert x\rvert \lt 1$.

Then I know that $$\tan^{-1}(x)=\sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{2n+1}$$ only for $\lvert x \rvert \lt 1$.

My professor asks me:

How do I know that $$\lim_{x\to\ 1}\sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{2n+1} = \sum_{n=0}^{\infty} \lim_{x\to\ 1}\frac{(-1)^nx^{2n+1}}{2n+1}\;?$$

In general, once the radius of convergence has been found for a power series, I think I can rest assured the series obtained by differentiating or integrating term by term has the same radius of convergence. What more do I need to consider to talk about the points on the radius?

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  • $\begingroup$ It's a lot more subtle, because the symbols don't interchange if you replace $\lim_{x \to 1}$ with $\lim_{x \to -1}$. $\endgroup$ – user296602 Apr 16 '18 at 20:28
  • $\begingroup$ Funnily enough, my professor also describes this as "subtle". Where should I begin reading? $\endgroup$ – Jungleshrimp Apr 16 '18 at 20:32
  • $\begingroup$ Have you heard about Abel, but not this one killed by Cain? $\endgroup$ – Przemysław Scherwentke Apr 16 '18 at 20:32
  • $\begingroup$ I have heard "of" Abel $\endgroup$ – Jungleshrimp Apr 16 '18 at 20:34
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    $\begingroup$ en.wikipedia.org/wiki/Abel%27s_theorem $\endgroup$ – Chappers Apr 16 '18 at 20:35

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