How do you find a power series converging to f(x)=cossqrtx and determine the radius of convergence?
1 Answer
Aug 7, 2017
f(x) = 1-(x)/(2!) + (x^2)/(4!) - (x^3)/(6!) + ...
This converges
Explanation:
First let us consider the well known Maclaurin series for
cos x = 1-(x^2)/(2!) + (x^4)/(4!) - (x^6)/(6!) + ...
And this series converges
So, now replace
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
