Binomial Series — Question 2

*Question 2 Treat $(1+x)^{-1}$ as a generalized binomial series. Derive its coefficients, connect the result with a geometric series, and determine the exact interval of convergence.

*Question 2 – Solution

For $n\ge0$, $$\binom{-1}{n}=\frac{(-1)(-2)\cdots(-n)}{n!}=(-1)^n.$$ Thus $$\boxed{(1+x)^{-1}=\sum_{n=0}^{\infty}(-1)^n x^n=1-x+x^2-x^3+\cdots}.$$

This is exactly $$\frac1{1-(-x)}=\sum_{n=0}^{\infty}(-x)^n,$$ whose common ratio is $-x$. It converges when $|-x|<1$, so $R=1$.

At $x=1$, terms are $(-1)^n$; at $x=-1$, terms are $1$. Neither tends to zero.

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