Evaluating Indefinite Integral as Power Series: Find Radius of Convergence

AI Thread Summary
The discussion focuses on evaluating the indefinite integral of (x - arctan(x))/x^3 and finding its radius of convergence. Participants suggest separating the integral into two parts for easier handling and recommend integrating the first part directly. The power series for arctan(x) is identified, and advice is given to divide its terms by x^3 before integrating term by term. There is also mention of using integration by parts for the new series and considering methods to determine the radius of convergence. Overall, the conversation emphasizes breaking down the problem and applying known series techniques to solve it.
jaidon
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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
 
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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.
 
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?
 
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).
 

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