Power series for integral of (x - arctan(x))/x³

[grothem]

Homework Statement

Evaluate the indefinite integral as a power series and find the radius of convergence

$$\int\frac{x-arctan(x)}{x^3}$$


I have no idea where to start here. Should I just integrate it first?

 

[Dick]

Sure, you could do that. But, I think what they want to do is expand arctan(x) as a power series around 0 and then integrate.

 

[grothem]

ok. So arctan(x) = $$\int\frac{1}{1+x^2}$$
= $$\int\sum (x^(2*n))$$
= $$\sum\frac{x^(2(n+1)}{2(n+1)}$$

is this what you mean?

 

[Dick]

That's one way to get a series for arctan, yes. But you forgot a (-1)^n factor. The expansion of 1/(1-x) has all plus signs. 1/(1+x) doesn't.