How do you find a power series converging to #f(x)=int t^-1 sintdt# from [0,x] and determine the radius of convergence?

Firstly a general disclaimer, With infinite sums there are some subtleties involved that we need to be careful with when differentiating and integrating term by term. You will not always get the result that you expect. But that is beyond the scope of this question!

The Maclaurin Series for sint is

#sint=t-t^3/(3!)+t^5/(5!)-t^7/(7!)+t^9/(9!)-t^11/(11!)...#

And so we can represent the power series of the integrand as follows;

# t^-1sint = sint/t #
# " " = {t-t^3/(3!)+t^5/(5!)-t^7/(7!)+t^9/(9!)-t^11/(11!)... }/t#
# " " = 1-t^2/(3!)+t^4/(5!)-t^6/(7!)+t^8/(9!)-t^10/(11!)... #

And so we can now form the power series for f(x)#:

# f(x) = int_0^x \ t^-1 \ sint \ dt #
# " " = int_0^x \ {1-t^2/(3!)+t^4/(5!)-t^6/(7!)+t^8/(9!)-t^10/(11!)...} \ dt #

Integrating term by term gives us:

# f(x) = [t-t^3/(3*3!)+t^5/(5*5!)-t^7/(7*7!)+t^9/(9*9!)-t^11/(11*11!)+...]_0^x #
# " " = x-x^3/(3*3!)+x^5/(5*5!)-x^7/(7*7!)+x^9/(9*9!)-x^11/(11*11!) +...#
# " " = x^1/(1*1!)-x^3/(3*3!)+x^5/(5*5!)-x^7/(7*7!)+x^9/(9*9!)-x^11/(11*11!) +...#
# " " = sum_(r=1)^oo \ (-1)^(r+1)x^(2r-1)/((2r-1)*(2r-1)!)#