how to remove the divergent term from the summation?

Question

I am doing the summation over two indices: m and n
Suppose I want to remove those terms from the summation which has m==n, what is the command for that? Because at m==n, terms are divergent.

I have already tried to do the summation in two cases:
m less than n and m>n.
But is there any other method to do it in one step?

Answer 1

Try also this:

    Sum[f[n, m] /. f[__] /; m == n -> 0, {n, 1, 3}, {m, 1, 3}]

(*   f[1, 2] + f[1, 3] + f[2, 1] + f[2, 3] + f[3, 1] + f[3, 2]  *)

The advantage is that if f[n,n]is divergent, Mma does not complain.

Answer 2

if the expression is not evaluated inside the Sum then you can try:

DeleteCases[Sum[f[n, m], {n, 1, 3}, {m, 1, 3}], f[i_, i_]]
(*f[1, 2] + f[1, 3] + f[2, 1] + f[2, 3] + f[3, 1] + f[3, 2]*)

if the expression is evaluated then try:

ind = DeleteCases[Tuples[Range[3], 2], {i_, i_}];
Total[f[#, #2] & @@@ ind]
(*f[1, 2] + f[1, 3] + f[2, 1] + f[2, 3] + f[3, 1] + f[3, 2]*)

Answer 3

Sum[f[n, m], {n, 3}, {m, n + 1, 3}] +  Sum[f[n, m], {n, 3}, {m, 1, n - 1}]