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Representing a function as a power series

by grothem
Tags: function, power, representing, series
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grothem
#1
Apr27-08, 09:49 PM
P: 23
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
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Dick
#2
Apr27-08, 10:49 PM
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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
#3
Apr28-08, 11:56 AM
P: 23
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
#4
Apr28-08, 01:41 PM
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Representing a function as a power series

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.


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