Introduction to Probability · Problem 12.2.2

Random Walks — Grinstead & Snell

Mathematics — Probability · gamblers-ruin-taylor-series

Problem

Show that the Taylor series expansion for 1−x\sqrt{1-x} is 1−x=∑n=0∞(1/2n)xn,\sqrt{1-x} = \sum_{n=0}^{\infty} \binom{1/2}{n} x^n, where the binomial coefficient (1/2n)\binom{1/2}{n} is (1/2n)=(1/2)(1/2−1)⋯(1/2−n+1)n!.\binom{1/2}{n} = \frac{(1/2)(1/2 - 1) \cdots (1/2 - n + 1)}{n!}.

Related theory

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