Question

How do you find the maclaurin series expansion of `cos(x^3)`?

Answer 1

By far the best way is to use the well-known Maclaurin series for `cos(x)` and replace all the `x's` with `x^3`'s.

Explanation:

We know that `cos(x)=1-x^2/(2!)+x^4/(4!)-x^6/(6!)+cdots` for all `x`. Therefore, `cos(x^3)=1-x^6/(2!)+x^12/(4!)-x^18/(6!)+cdots` for all `x`.

It's very unpleasant, but if you let `f(x)=cos(x^3)`, you could also try using:

`f(0)+f'(0)x+(f''(0))/(2!)x^2+(f'''(0))/(3!)+(f''''(0))/(4!)+cdots`