Define \(\displaystyle f(x) = \sum_{n=0}^{\infty} a_{n} x^n\) and \(\displaystyle A(n) = \sum_{r=0}^{n} a_{r}\)

Give that \(\displaystyle \lim_{n \to \infty} \dfrac{A(n)}{n^r} = \alpha \)

Prove That

\[ \lim_{x \to 1^{-}} (1-x)^r f(x) = \alpha \ \Gamma(1+r) \]

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