representing a function as a power series

grothem

1. The problem statement, all variables and given/known data
Evaluate the indefinite integral as a power series and find the radius of convergence

[tex]\int\frac{x-arctan(x)}{x^3}[/tex]


I have no idea where to start here. Should I just integrate it first?
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

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) = [tex]\int\frac{1}{1+x^2}[/tex]
= [tex]\int\sum (x^(2*n))[/tex]
= [tex]\sum\frac{x^(2(n+1)}{2(n+1)}[/tex]

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.