Infinite sum limit

Discussion in 'Analysis and Topology' started by Mike, Dec 2, 2017.

  1. Mike

    Mike

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    I performed an inverse Laplace transform (probably incorrectly) and found a function which is very strange when I plot it. The function is 1/2 + Sum{cos(2n pi x) - cos((2n+1)pi x), n=1, infinity}. As n increases the plot oscillates rapidly starting at x = 0 and as x -> 1 it looks like -delta(x) just before going to delta(x) at x = 1. What tricks can be used to show this is a Fourier expansion of something reasonable? Or at least some closed form of something unreasonable!

    Thanks.
    Mike
     
    Mike, Dec 2, 2017
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  2. Mike

    HallsofIvy

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    Trig identity: cos(a)- cos(b)= 2 sin((a+ b)/2) sin((a- b)/2).

    Here a= 2npi x and b= (2n+ 1)pi x so a+ b= (4n+ 1)pi x and a- b= -pi x.
     
    HallsofIvy, Nov 14, 2021
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  3. Mike

    Mike

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    So sin(pi x) -> 0 as x -> 1 independent of n and sin((4n + 1) pi x) oscillates like crazy as x -> 1 as a function of n. I don't remember what I was doing any more because it's been a while, but that's a nice simple way to see it is pathological. Thanks!
     
    Mike, Nov 14, 2021
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