How do you find the power series for #f(x)=int t^2/(1+t^2)dt# from [0,x] and determine its radius of convergence?
2 Answers
with radius of convergence
Explanation:
Write the integral as:
Using the linearity of integrals:
Now consider the integrand function: it is in the form of the sum of a geometric series of ratio
and the series is convergent for
Within this interval we can therefore integrate term by term and we have:
Substituting this in the expression above we have:
Now note that the first term of the series is just
and we can conclude that:
with radius of convergence
Explanation:
You could have a go at this using the FTC, but it's gonna be easier just doing the integration and adding in some predetermined Maclaurin Series (I know cos it started doing it the other way and it's a real drag).
If we start with:
This is a trivial integral so I'll post the result:
The power series for
It follows that:
The ratio test confirms convergence at
If we go at it using the FTC , ie using this:
...then we can start to generate the terms from the fundamental definition/idea, which is that:
So we start with:
#f(0) = int_0^0 t^2/(1+t^2)dt = 0 #
After that, using the FTC and differentiation:

#f' (x) = x^2/(1+x^2)# 
#f'' (x) = d/dx( x^2/(1+x^2))#
And you can loop through that and it's really, really boring and longwinded.