Binomial Series — Question 3

*Question 3 Find the first four nonzero terms of $(1+x)^{-1/2}$ using generalized binomial coefficients. Explain the signs and determine the exact interval of convergence.

*Question 3 – Solution

$$\binom{-1/2}{1}=-\frac12,\quad \binom{-1/2}{2}=\frac38,\quad \binom{-1/2}{3}=-\frac5{16}.$$ Every factor in the numerator is negative, so the coefficient sign is $(-1)^n$. Hence $$\boxed{(1+x)^{-1/2}=1-\frac{x}{2}+\frac{3x^2}{8}-\frac{5x^3}{16}+\frac{35x^4}{128}-\cdots}.$$

The generalized binomial series has $R=1$.

The coefficient magnitudes satisfy $$\left|\binom{-1/2}{n}\right|=\frac1{4^n}\binom{2n}{n}\sim\frac1{\sqrt{\pi n}}.$$ At $x=1$ the signs alternate, so the series converges conditionally. At $x=-1$, all terms are positive and comparable with $1/\sqrt n$, so it diverges.

The endpoint magnitudes $b_n=4^{-n}\binom{2n}{n}$ decrease, since $b_{n+1}/b_n=(2n+1)/(2n+2)<1$, and tend to zero. This verifies the alternating-test hypotheses.

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