Question

How do you find a power series converging to `f(x)=cossqrtx` and determine the radius of convergence?

Answer 1

`f(x) = 1-(x)/(2!) + (x^2)/(4!) - (x^3)/(6!) + ...`

This converges `AA x in RR`

Explanation:

First let us consider the well known Maclaurin series for `cos x`. We could derive this from first principles, but is is rarely required to do this and is quoted on most examination formula books:

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

And this series converges `AA x in RR`

So, now replace `x` by `sqrt(x)` and we get:

`f(x) = cos sqrt(x)`
`" " = cos(x^(1/2))`
`" " = 1-((x^(1/2))^2)/(2!) + ((x^(1/2))^4)/(4!) - ((x^(1/2))^6)/(6!) + ...`
`" " = 1-(x)/(2!) + (x^2)/(4!) - (x^3)/(6!) + ...`

Again, this converges `AA x in RR`