Question

Given 1- cosx, how do you find the Taylor polynomial?

Answer 1

We should know that:

`cosx=sum_(n=0)^oo(-1)^nx^(2n)/((2n)!)=1-x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)+...`

So

`1-cosx=1-(1-x^2/(2!)+x^4/(4!)-x^6/(6!)+x^8/(8!)+...)`

`color(white)(1-cosx)=x^2/(2!)-x^4/(4!)+x^6/(6!)-x^8/(8!)+...`

Noting this still alternates and that the factorials and powers increase by `2` in tandem:

`1-cosx=sum_(n=0)^oo(-1)^nx^(2n+2)/((2n+2)!)`

Which we could also write as:

`1-cosx=sum_(n=1)^oo(-1)^(n-1)x^(2n)/((2n)!)`