Consider the Binomial series
\begin{align} (1+x)^\alpha =\sum^{\infty}_{k=0}{\alpha\choose k}x^k,\;\text{for }\;x\in \Bbb{R}\;\text{and}\;\alpha\in\Bbb{R} \end{align} I want to show that it converges uniformly on $\Bbb{R}$.
However, it can be shown by D'Alembert's Ratio test that the following series converges absolutely \begin{align} F(\alpha) =\sum^{\infty}_{k=0}{\alpha\choose k}x^k,\;\text{for fixed}\;|x|<1\;\text{and}\;\alpha\in\Bbb{R} \end{align} since \begin{align} \lim\limits_{k\to\infty}\left|\dfrac{^\alpha C_{k+1}}{^\alpha C_{k}}\right|=\lim\limits_{k\to\infty}\left|\dfrac{\alpha-k}{k+1}\right|=|x|<1,\;\text{for fixed}\;|x|<1\;\text{and}\;\alpha\in\Bbb{R} \end{align} QUESTION: How do I get uniform convergence of $F$ on $\Bbb{R}$ from there? Alternatively, if there's any other way of showing that it converges uniformly on $\Bbb{R}$, I will appreciate.
A series of the type $\sum a_k x^{k}$ cannot converge uniformly on $\mathbb R$ unless $a_k=0$ for all but finite number of $k$'s. (This is because the general term does not tend to $0$ uniformly).