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I'm trying to find how this series is continuous by proving its infinitely differentiable. In my books, i came across the internal of convergence and the radius of convergence which should help me figure this out, but I cant seem to wrap my head around the concept of the power series to begin with, and my research generally associates this series with the taylor series; is it the same thing? can i still prove that this function is continuous everywhere through this method?

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  • $\begingroup$ Proving a function infinitely differentiable is usually much harder than proving it is continuous. $\endgroup$ – GEdgar Feb 10 '18 at 14:39
  • $\begingroup$ idk i tried doing the latter, but it just didnt come easily to me. this seems comparatively easier. $\endgroup$ – Cookiedough Feb 10 '18 at 15:44
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Hint

$$a^x=e^{x\ln(a)},\quad a>0.$$

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