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I'd like to prove the following proposition:

The power series $\sum_{j=0}^{\infty} a_j(x - c)^j$ and the series $\sum_{j=0}^{\infty} \frac{a_j}{j+1}(x - c)^{j+1}$ obtained from term by term integration have the same radius of convergence, and the function $F$ defined by $F(x) = \sum_{j=0}^{\infty} \frac{a_j}{j+1}(x - c)^{j+1}$ on the common interval of convergence satisfies $F'(x) = \sum_{j=0}^{\infty} a_j(x - c)^j = f(x)$.

Does this occur as a result of $\sum_{j=0}^{\infty} a_j(x - c)^j$ and $\sum_{j=0}^{\infty} \frac{a_j}{j+1}(x - c)^{j+1}$ being real analytic? (Are they real analytic?)

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    Cauchy-Hadamard theorem.2012-03-16

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