Can we apply binomial theorem for $\quad(a+b)^\ell\quad$ if $\ell\;$ irrational.

Question

Let be$\quad a,b\;\in\mathbb R\quad, \ell\;\in\mathbb {(R\backslash Q)} \quad $ ($\ell:$irrational)

Can we apply binomial theorem for $\quad(a+b)^\ell$

Answer 1

Yes: $$ (a+b)^{l}=a^{l}\sum_{k=0}^{\infty}\binom{l}{k}\bigg(\frac{b}{a}\bigg)^k $$ Binomial coefficients are calculated as the regular ones. $$ \binom{\alpha}{k}=\alpha \cdot (\alpha-1) \cdots (\alpha-k+1) \cdot \frac{1}{k!} $$

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