Question

Define a function sinc(x) (pronounced "sink of x") by: sinc(x)= {sin(x)/x if x is not 0 {1 if x = 0 Use this list of Basic Taylor Series to find the Taylor Series for f(x) = (sinc(x)) based at 0. a.Give your answer using summation notation. b.Give the interval on which the series converges.

Answer 1

You probably know that

$\sin x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}$

Then

$\mathrm{sinc}\,x=\displaystyle\frac1x\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}=\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}$

when $x\neq0$, and 1 when $x=0$.

By the ratio test, the series converges if the following limit is less than 1:

$\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{2n+2}}{(2n+3)!}}{\frac{(-1)^nx^{2n}}{(2n+1)!}}\right|=|x^2|\lim_{n\to\infty}\frac{(2n+1)!}{(2n+3)!}$

The limit is 0, so the series converges for all $x$.