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Max of sum of sines

  1. Apr 23, 2014 #1

    Consider the function

    [itex] \frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}, \quad 0 \leq x \leq \pi/2 [/itex].

    If [itex] n [/itex] is odd this function attains its largest value, [itex] \sum_{k=1}^m k^n [/itex] at [itex] x=0 [/itex]. But what about if [itex] n [/itex] is even? Where does the maximum occur and what value does it take?

    Any help is much appreciated. Thank you!
    Last edited: Apr 23, 2014
  2. jcsd
  3. Apr 23, 2014 #2

    Simon Bridge

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    Please make some attempt at the problem when you ask for help.
    i.e. how would you normally go about finding the maximum?
    Can you prove the statement about when m is odd?
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