Let $\alpha\in\mathbb{R}$. Prove that, for all $x \in [0, 1)$ we have $(1 + x)^\alpha=1+{\alpha x\over 1!}+{\alpha(\alpha-1)x^2\over 2!}+\cdots+{\alpha(\alpha-1)\cdots(\alpha-k+1)x^k\over k!}+\cdots$.

I'm not sure how to do this. Maybe Taylor's Remainder Theorem. Any hints or solutions are greatly appreciated.

  • $\begingroup$ Have you tried to see what Taylor's Formula gives? $\endgroup$ – robjohn Apr 23 '15 at 21:48
  • $\begingroup$ Since it is an approximation of the function around zero you use Maclaurin series. $\endgroup$ – Superman Apr 23 '15 at 21:54
  • $\begingroup$ en.wikipedia.org/wiki/Binomial_series $\endgroup$ – vadim123 Apr 23 '15 at 21:55
  • $\begingroup$ Is that all I need to do? Use Taylor Series? $\endgroup$ – Turd Ferguson Apr 24 '15 at 2:54

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