Consider the sum
where >0 for all N. Since all the terms are positive, the value of S must be greater than the value of the largest term, , and less than the product of the number of terms and the value of the largest term. So we have
Taking logarithm gives
In statistical mechanics often  will be : see Big O notation.
Here we have
For large M,  is negligible with respect to M itself, and so we can see that ln S is bounded from above and below by , and so