In fact, Euler already knew this. Intuitively one can simply try the well-known closed form for geometric series, that is
for (at your own risk!), to find the solution:
Alternatively, you can use Riemann’s Zeta function
which, with , results after heavy rewriting in again.
For a different perspective, see Bill Gosper in HAKMEM 154, where he figures:
By this strategy, consider the universe, or, more precisely, algebra:
now add X to itself;
therefore algebra is run on a machine (the universe) which is twos-complement.