- #1
Whistlekins
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Homework Statement
Prove that the series [itex]\sum_{n=0}^\infty(-1)^n\frac{x^{2n+1}}{2n+1}[/itex] is well-defined and differentiable on (-1,1).
Homework Equations
The Attempt at a Solution
I know that the function is the series expansion of arctan(x), but that it not we are showing here (however it asks in a later question to show that it is). I don't know what it means by "well-defined", but I'm going to guess it means continuous and convergent on its domain. I am guessing that I should use the Uniform Cauchy Criterion to show that it converges uniformly, and thus showing that it is differentiable.
But I'm not sure how to show that for all ε > 0, there exists and N ≥ 1 such that, for all n, p ≥ N and all x in (-1,1), |f_n(x) - f_p(x)| < ε
Also, how would I show that it is actually the series expansion for arctan(x)?
A nudge in the right direction would be greatly appreciated.