1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Series of functions

  1. May 29, 2013 #1
    1. The problem statement, all variables and given/known data

    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).

    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. May 29, 2013 #2
    That sounds like a good plan. Can you see when |f_n(x) - f_p(x)| would be maximized? You can bound this from above by choosing a value for x that gives the largest difference.

    You can probably show that their derivatives are equal.
     
  4. May 29, 2013 #3
    Can you expand on what you mean by maximized? Would that happen when n = 0 and p -> ∞?
     
  5. May 29, 2013 #4
    I mean choose value for n, then pick a p that maximizes the difference for arbitrary x and then choose x to find the absolute upper bound for a given n. And notice that the series is alternating, so n=0 and p→∞ is not the maximum.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Series of functions
  1. Functions of Series (Replies: 16)

Loading...