For any real number \(\alpha\) and nonnegative integer \(n\text{,}\) define \(\binom{\alpha}{n}=\alpha (\alpha-1)\cdots (\alpha-n+1)/n!\text{.}\) Show that the radius of convergence of the power series \(\sum_{n=0}^{\infty} \binom{\alpha}{n} x^{n}\) equals \(1\text{.}\) Then show that \((1+x)^{\alpha}= \sum_{n=0}^{\infty} \binom{\alpha}{n} x^{n}\) for \(|x|\lt 1\text{.}\)
Suppose that the power series \(\sum_{n=0}^{\infty} a_{n}x^{n}\) has radius of convergence equal to \(1\text{,}\) that each \(a_{n}\ge 0\) and that \(\sum_{n=0}^{\infty} a_{n}=\infty\text{.}\) Show that \(\lim_{x\to 1-} \sum_{n=0}^{\infty} a_{n}x^{n}=\infty\text{.}\)
Suppose that the power series \(f(x)=\sum_{n=0}^{\infty} a_{n}x^{n}\) has a positive radius of convergence, and \(g(y)=\sum_{n=0}^{\infty} b_{n} (y-a_{0})^{n}\) also has a positive radius of convergence \(R\text{.}\) Prove that for all \(x\) such that \(\sum_{n=1}^{\infty} |a_{n}x^{n}| \lt R\text{,}\) the composite function \(g\circ f(x)\) has a convergent power series expansion of the form \(\sum_{n=0}^{\infty} c_{n}x^{n}\text{,}\) where \(c_{0}=b_{0}\) and \(c_{n}=\sum_{k=0}^{n} b_{k} a_{n}(k)\) for positive integer \(n\text{,}\) with \(a_{n}(k)\) defined from the relation \((\sum_{k=1}^{\infty} a_{k}x^{k})^{n}
=\sum_{k=1}^{\infty} a_{k}(n) x^{k}\text{.}\)
