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

1 Answer
Steve M
Aug 7, 2017

# 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#

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