# Can a power series with rational cffs. that sum to irrational lim evaluate to rational lim at non-zero rational point?

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!

• You should use "$\mathbb Q$" for the rationals. The symbol "$\mathbb R$" pretty universally already means "real numbers".
– MPW
Aug 13, 2019 at 19:45
• @JoséCarlosSantos : I think OP is using $\mathbb R$ for "rational numbers" -- yikes!
– MPW
Aug 13, 2019 at 19:46
• yep my bad, that is embarrassing >.> Aug 13, 2019 at 19:47

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 $$\sqrt{1/4}=1/2$$.