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I have a power series $\displaystyle \sum \limits_{n \in \mathbb{N}} a_{n}x^{n}$ whose radius of convergence is equal to $4$. For all $x \in ]-4,4[$, let $f(x) = \displaystyle \sum_{n=0}^{+\infty} a_{n}x^{n}$. I am asked to prove that $f$ has a right derivative at $x=0$. The answer seems obvious to me since we know that $f$ is even infinitely differentiable on $]-4,4[$. Is that a suitable answer or am I missing something ?

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If the radius of convergence is $4$, therefore $f$ is derivable on $]-4,4[$ and so $f$ has a right derivate at $x=0$ (because $f$ is derivable at $x=0$).

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