Binomial series

Example

Let $\alpha$ be a real number. For each natural number $k$ , let $\binom{\alpha}{k} = \frac{\alpha (\alpha - 1) \cdots (\alpha - k + 1)}{k!}.$ The binomial series is the power series $\sum_{k=0}^{\infty} \binom{\alpha}{k} x^k$ . For $|x| < 1$ its sum is $(1 + x)^{\alpha}$ .

Remarks

If $\alpha$ is a natural number, then $\binom{\alpha}{k} = 0$ for $k > \alpha$ . The series is then the binomial formula for $(1 + x)^{\alpha}$ . Otherwise the radius of convergence is 1. For $\alpha = -1$ it is the geometric series $\sum (-1)^k x^k = 1/(1 + x)$ . For $\alpha = 1/2$ it gives $\sqrt{1 + x} = 1 + x/2 - x^2/8 + \cdots$ .

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Sequence definition
Real sequence definition
Series definition
Power series definition
Binomial series example

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Radius of convergence of the binomial series

Tag	`0HB0XKDXHH`
URI	`m2://0HB0XKDXHH`
Kind	example
Writing	W2 usable
MSC	30B10
Instance of	Power series
Uses	Power series
Same as	Wikidata Q833480 Wikipedia Binomial series

Binomial series

Example

Remarks

Reading path

Where this leads

Further reading

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