I am facing the problem in finding out the root of the equation. Feedback is appreciated.
The cover posting for a notebook should include a summary of the problem and a clear question.
In this case, both of the Sums add to zero so your equation is 0/0 == 0.25. It may be that the limits on the sums are not correct.
Please try evaluating pieces of the equation separately to find where it goes astray.
Moderation Team
Perhaps use Simplify to handle the denominators.
In[1]:= Clear[m, k, l, a, b, A, B];
simp = Simplify[1/Sum[Binomial[m-k, l]*(-1)^l*Gamma[k+a]/( b + B + k/A^2)^(k+a), {l, 0, m-k}]*
Sum[Binomial[m-k, l]*(-1)^l*Gamma[k+a]/(1/L^2 + b + B + k/A^2)^(k+a), {l, 0, m-k}]];
m = 35; k = 32; \[Lambda] = 1; a = 2; b = 2; u = RandomReal[UniformDistribution[{0, 1}], m];
t = ((-1*\[Lambda])/Log[u])^(1/2); s = t; w = Sort[s]; st = Take[w, k]; B = Total[st^-2]; A = Last[st];
FindRoot[simp == 0.25, {L, 1}]
FindRoot[simp == 0.75, {L, 1}]
Plot[{simp - 0.25, simp - 0.75}, {L, 0, 3}]
Out[4]= {L -> 0.72545}
Out[5]= {L -> 1.60547}
Please check this carefully. Thank you