A post from a few months ago gave a proof that In today’s post I’d like to prove a general symmetry formula for Euler sums like this one. Define

and

(Notice the upper index on the second sum is , not like we used in the previous post.) The formula we will be proving is

.

We will show this by proving a more precise result:

In other words, the symmetry result is true in both the finite and infinite versions; there’s no error term when you truncate the sum at a fixed *N*.

Here we go:

, by swapping the order of summation on the first sum

, by relabeling variables on the first sum

Taking the limit as , we get

.

(This argument also appears in my post on the evaluation of a triple Euler sum on math.SE.)

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