Difference between revisions of "Binomial series"
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(Created page with "==Theorem== The following formula holds for any $\alpha \in \mathbb{C}$: $$(1+x)^{\alpha} = \displaystyle\sum_{k=1}^{\infty} {\alpha \choose k} x^k,$$ where ${\alpha \choose k...") |
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The following formula holds for any $\alpha \in \mathbb{C}$: | The following formula holds for any $\alpha \in \mathbb{C}$: | ||
$$(1+x)^{\alpha} = \displaystyle\sum_{k=1}^{\infty} {\alpha \choose k} x^k,$$ | $$(1+x)^{\alpha} = \displaystyle\sum_{k=1}^{\infty} {\alpha \choose k} x^k,$$ | ||
| − | where ${\alpha \choose k}$ denotes the [[binomial coefficient]]. | + | where $\displaystyle{\alpha \choose k}$ denotes the [[binomial coefficient]]. |
==Proof== | ==Proof== | ||
Latest revision as of 12:22, 11 August 2016
Theorem
The following formula holds for any $\alpha \in \mathbb{C}$: $$(1+x)^{\alpha} = \displaystyle\sum_{k=1}^{\infty} {\alpha \choose k} x^k,$$ where $\displaystyle{\alpha \choose k}$ denotes the binomial coefficient.
