Hi Reza,
The sampling you have chosen in Table is not fine enough to reveal the structure around 2.5. The spike occurs at r = 2.51
NeAB // MaximalBy[Last]
(* {{251/100, 0.512555}} *)
To examine that area in more detail
f[r_] := -2 N[
Sum[1/
12 (2 + 2 n - Sqrt[2 (5 + 2 n (2 + n) + 4 Cosh[2 r] + Cosh[4 r])]) Sech[r]^4 Tanh[r]^(2 n),
{n, 0, \[Infinity]}]]
This takes ~6 min on my laptop
Plot[f[x], {x, 2.5, 2.53}, ImageSize -> 600]
The function has a discontinuity around r = 2.514
