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((-1)^nx^(2n+1))/((2n+1)!)`

Then

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

when
`x\\eq0`, 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|(((-1)^(n+1)x^(2n+2))/((2n+3)!))/(((-1)^nx^(2n))/((2n+1)!))\right|=|x^2|\lim_(n\to\infty)((2n+1)!)/((2n+3)!)`

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