Question

How do you find the Maclaurin Series for `f(x)=cos(5x^2)`?

Answer 1

The Maclaurin series for `cos(x)` is well known:

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

So:

`cos(5x^2)=sum_(n=0)^oo(-1)^n/((2n)!)(5x^2)^(2n)=1-(5x^2)^2/(2!)+(5x^2)^4/(4!)-(5x^2)^6/(6!)+...`

`color(white)(cos(5x^2))=sum_(n=0)^oo((-1)^n5^(2n))/((2n)!)x^(4n)`