The sum we want is
where is the n-th triangular number, with a repeating sign pattern (+, -, -, +). We can rewrite this sum as
For convenience, I'll use the abbreviations
for m ∈ {1, 2, 3, …, 7}, as well as the well-known series
We want to find .
Consider the periodic function on the interval [0, 1], which has the Fourier expansion
That is, since f(x) is even,
where
(See attached for a plot of f(x) along with its Fourier expansion up to order n = 10.)
Expand the Fourier series to get sums resembling the -s :
which reduces to the identity
Evaluating both sides at x for x ∈ {1/8, 3/8, 5/8, 7/8} and solving the system of equations yields the dependent solution
It turns out that
so we're done, and the sum's value is .
