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Let $f(z)=\sum a_nz^n$ be a power series with radius of convergence $R.$ Now can i say that there is at least one point on the circle of convergence $|z|=R$ at which the power series will not converge? I am thinking so because if the power series is convergent at all points of the the circle of convergence then $f$ will be analytic at $|z|=R$ and radius of convergence will increase and which is not possible. Am i right? Please suggest me. Thanks a lot.

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You can consider $a_n=1/n^2$. The power series has radius of convergence 1 but it converges at all points of the boundary of unit circle.

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  • $\begingroup$ So does $\sum z^n/n^2$ not have a pole within $|z|\leq 1$? What is the pole structure of this function? $\endgroup$ – ziggurism Nov 4 '15 at 13:33
  • $\begingroup$ According to this wolfram page, dilogarithm does not have any poles. But I'm confused. Isn't a meromorphic function analytic in any domain where it does not have any poles? Shouldn't this make the dilogarithm anaylitic on $\mathbb{C}$ and therefore its Taylor series converges on $\mathbb{C}$? $\endgroup$ – ziggurism Nov 4 '15 at 13:55
  • $\begingroup$ This question seems to contain an answer to my question, but I didn't quite understand it. Maybe I should start a new question. $\endgroup$ – ziggurism Nov 4 '15 at 15:10

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