## Question

###### Use the following steps to prove $(17) .$ (a) Let $g(x)=\Sigma_{n=0}^{\infty}\left(\frac{k}{n}\right) x^{n} .$ Differentiate this series to show that $$g^{\prime}(x)=\frac{k g(x)}{1+x} \quad-1< x<1$$ (b) Let $h(x)=(1+x)^{-k} g(x)$ and show that $h^{\prime}(x)=0$ (c) Deduce that $g(x)=(1+x)^{k}.$

Use the following steps to prove $(17) .$ (a) Let $g(x)=\Sigma_{n=0}^{\infty}\left(\frac{k}{n}\right) x^{n} .$ Differentiate this series to show that $$g^{\prime}(x)=\frac{k g(x)}{1+x} \quad-1< x<1$$ (b) Let $h(x)=(1+x)^{-k} g(x)$ and show that $h^{\prime}(x)=0$ (c) Deduce that $g(x)=(1+x)^{k}.$

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