Evaluating Indefinite Integral as Power Series: Find Radius of Convergence

[jaidon]

Evaluate the indefinite integral as a power series and find radius of convergence. (i don't know how to type the integral and summations signs, sorry)


(integral sign) (x-tan^-1x)/x^3 dx. ( if you write this out it makes more sense)

i was able to find the power series of tan^-1x = x^(2n+1) (-1)^n/(2n+1).
i don't know how to continue on with this. all we have learned is to use the power series of the geometric series 1/(1-x), and some integration/differentiation methods.

i am rather confused on the whole topic, so if anyone has any ideas, the simplest explanations would be greatly appreciated. thanks

 

[dextercioby]

So your integral is
$$\int \frac{x-\arctan x}{x^{3}} dx$$

??Okay,for term by term integration of it,separate it into 2 integrals...Though it's not really helpful for the convergence part...

Daniel.

 

[jaidon]

that is the integral, thanks, but i am not sure what to do after pulling it apart into two integrals. quite honestly, i am puzzled by this whole topic. any advice?

 

[dextercioby]

1.Pull apart into integrals.
2.Integrate the first.It's elementary.
3.Write the series expansion of "artan" and divide its terms by x^{3}.
4.Integrate by parts eery term of the new series...
5.Think of a way to get the convergence radius.

Daniel.

P.S.It can be done exactly (find the antiderivative).