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Someone asked a question here about why the Taylor series of $\log(1+x)$ diverges:

Why does the taylor series of $\ln (1 + x)$ only approximate it for $-1<x \le 1$?

I have a similar question: why does the $\operatorname{Sech}(x)$ Taylor series diverge at $\pi/2$?

The answer in the $\log$ question dealt with the fact that the only symmetric interval that could be defined for $\log$ is from $-1$ to $1$ as $\log$ is only defined from $(-1, \infty)$. $\operatorname{Sech}(x)$ is defined everywhere and it doesn't have this problem.

Thanks for all your help.

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    $\begingroup$ The reason is the same as for the Taylor series of $1/(1+x^2)$ that only converges for $-1<x<1$. $\endgroup$
    – egreg
    Commented Jul 12, 2016 at 16:02
  • $\begingroup$ Radius of convergence is answered. But how about convergence at the point $\pi/2$, which is on the boundary of the region of convergence? $\endgroup$
    – GEdgar
    Commented Jul 12, 2016 at 16:16
  • $\begingroup$ the major contribution of comples analysis is that there is an equivalence between "a power series, analytic on a disk" and "no singularity on that disk", i.e. $f(z) = \sum_{n=0}^\infty c_n z^n$ always has a singularity on the boundary of its disk of convergence, and conversely, it doesn't converge past its largest singularity-free disk $\endgroup$
    – reuns
    Commented Jul 12, 2016 at 16:46

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Easiest reason is in complex analysis. $\mathrm{sech}(x)$ has a pole at $i\pi/2$, so the radius of convergenge cannot extend beyond that point. And moreover, there are no singularities strictly closer to $0$, so the radius of convergence is exactly $\pi/2$.

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