Binomial Series — Question 1

*Question 1 Use the generalized binomial theorem to find the first four nonzero terms of $(1+x)^{1/2}$. State the radius and determine convergence at both endpoints.

*Question 1 – Solution

$$(1+x)^\alpha=\sum_{n=0}^{\infty}\binom{\alpha}{n}x^n,\qquad \binom{\alpha}{n}=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}.$$

$$\binom{1/2}{0}=1,\quad \binom{1/2}{1}=\frac12,\quad \binom{1/2}{2}=-\frac18,\quad \binom{1/2}{3}=\frac1{16}.$$ Therefore $$\boxed{\sqrt{1+x}=1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\cdots}.$$

The radius is $R=1$. The coefficient magnitudes are asymptotic to a constant times $n^{-3/2}$, so the series converges absolutely at both $x=1$ and $x=-1$.

The interval is $\boxed{[-1,1]}$.