Assume we have $f(x) = \sum a_n x^n, a_n \in \mathbb{Q}$, and convergent $f(1) \notin \mathbb{Q}$. Assuming $f(x)$ converges at some $f(x \in \mathbb{Q})$, is it possible for $f(x \in \mathbb{Q}) \in \mathbb{Q}$ for some non-zero $x$?
Thanks!
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$\begingroup$ You should use "$\mathbb Q$" for the rationals. The symbol "$\mathbb R$" pretty universally already means "real numbers". $\endgroup$– MPWAug 13, 2019 at 19:45
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$\begingroup$ @JoséCarlosSantos : I think OP is using $\mathbb R$ for "rational numbers" -- yikes! $\endgroup$– MPWAug 13, 2019 at 19:46
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$\begingroup$ yep my bad, that is embarrassing >.> $\endgroup$– NiklasAug 13, 2019 at 19:47
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Consider $$ \sqrt{1+\frac{x}{2}}=\sum_{k\ge0}\binom{1/2}{k}\frac{1}{2^k}x^k $$ that converges to $\sqrt{3/2}$ for $x=1$.
Take $x=-3/2$; then the series converges to $\sqrt{1/4}=1/2$.
