I am a researcher and encountered the following challenging function in my work:
$$f(S)=\sum_{k=1}^{S-1} \bigg[ \frac{1}{k^2(2k-1)^2(2k+1)^2} \bigg] \bigg( (2k-1)[-\ln(1+4S)+\ln(-1+4(S-k))] + (2k+1)[+\ln(-1+4S)-\ln(1+4(S-k))] \bigg)^2 $$
And I am only interested in the first term of the Taylor expansion of this function when $S\rightarrow+\infty$. Matlab simulations give me that it is equivalent to:
$$\frac{a}{S^4}\ , \quad a>0$$
In other words, simply a positive scalar divided by the parameter $S^4$.
Do you have any idea how to compute this term?
PS: I have tried to manipulate the expression such as to obtain a Riemann sum like in the previous post (Challenging Taylor expansion involving a summation of logarithms), but this seems more complicated to obtain.
Thank you so much.