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Problem in converting summation to integral

  1. Jun 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Thank you very much for helping me.

    I have to convert the following summation of a term from 1 to ∞ to a definite integral.

    Sum for k=1 to ∞: (2+Cosh[2k/x]) Csch^4 [k/x]

    I have already tried the rules for converting from different sources and websites, which is

    replace r/n by x
    repalce1/n by dx
    replace Ʃ by ∫

    but when i plot both the summation and the obtained integral, both are not same. Please tell me where i am doing wrong.

    Limit n->∞: Sum for k=1 to n: (2+Cosh[2k n/(x n)]) Csch^4 [k n/(x n)] (n/n)

    Since i cannot take "1/n" common, i multiplied and divided by "n".

    1/n->dΔ
    k/n->Δ

    it becomes,

    (2+Cosh[2Δn/x]) Csch^4 [Δn/x] dΔ limit: 1/n to 1

    It will converge for n~10, so we don't have to use "n->∞" and so the solution will not diverge.

    The Integral obtained is

    x Coth[1/x] Csch[1/x]^2 - x Coth[n/x] Csch[n/x]^2

    When i plot both, the above solution after integration and the actual sum are not the same. Please help me solving this.

    I am also including the mathematica file along with this post.
     

    Attached Files:

  2. jcsd
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