real analysis - Can someone give me an example of a function $f$ being analytic but its power series $sum_{n=0}^{infty} a_{n}x^{n}$ diverging for some $x$?

is it necessarily true that $f$ being analytic implies its power series converges for all $x$? I think that it cannot diverge; however, I'm not very good at coming up with counterexamples. Can someone please help me? I believe that no such $f$ exists.

Answer

If it is analytic for all complex values of $x$, then the power series will converge. However if you mean analytic for all real values of $x$, then the power series will not converge if there is a singularity for some non-real (complex) value. Simple example: $\frac{1}{1+x^2}$ with power series $\sum_{n=0}^\infty (-1)^nx^{2n}$.