Problem in converting summation to integral

[venki_k07]

Homework Statement

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.

 

[lippyka]

Homework Equations The equations I used were mentioned in the problem statement.The Attempt at a Solution I attempted the problem as mentioned in the problem statement, but the answer obtained is not correct.