Can someone check something for me in the computational sense?

Question

Here it is presented the sum which appeared in a recent mathematical competition at a local university: $$\sum_{j,k,l\geq0} \frac{1}{3^l\left(3^{j+k}+3^{k+l}+3^{l+j}\right)}$$ and it is said that the answer is $9/8$.

There also the answer is given which proves that the sum is $9/8$ but I am somehow disagreed with that approach of proving that the sum is $9/8$, and that disagreement is based on some pencil-and-paper calculations of mine that suggest me that this sum is not equal to $9/8$, in hope that those calculations are right.

So, I am not asking too much, only that someone calculates this sum for some cube constrained by $(0\leq j\leq m)$ and $(0\leq k\leq m)$ and $(0\leq l\leq m)$, where she/he can take that $m$ to be small enough in a sense that computer does not take much time to calculate the sum, but again, if my approach is right, then $9/8$ should be exceeded at some point.

I do not have any programming languages on a computer so I cannot do this task without your help.

Answer 1

// gcc threesum.c -o threesum.exe -std=c99 -Wall -O3

#include <math.h>
#include <stdio.h>

int main() {

  long double sum = 0;

  for (int total = 30; total >= 0; total--)
  for (int j = 0; j <= total; j++)
  for (int k = 0; k <= total - j; k++) {
    int l = total - j - k;

    long double t = 3.0;
    long double denom = pow(t, l)*( pow(t, j + k) + pow(t, k + l) + pow(t, l + j) );
    long double addend = 1.0 / denom;

    sum += addend;
  }

  printf("%f\n", (double)sum);
}


1.125000