Monthly Archives: February 2014

An unusually alternating sum

Problem: evaluate the following sum… $$\sum_{k=0}^{\infty}\dfrac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}$$ This is unusual because the $-1$ is raised to the $k (k+1)/2$ power, rather than the usual $k$th power. On the surface, this problem may look hopeless, but really, it is all about determining the pattern of odd and even numbers from the sequence $k (k+1)/2$, which turns out […]

Algebraically difficult integral

Well, some integrals are not all that hard to evaluate in principle. The one I am posting here should be an open and shut application of the residue theorem, using the unit circle as a contour. The form of the integrand, however, should give a little pause. It turns out that actually computing residues on […]

An integral involving a quadratic phase

This one was first posted on the site Integrals and Series and was brought to my attention on M.SE by Cody. My solution involves a contour integration, although the approach is far from trivial. Yet again, the solution boils down to finding a contour and a function to integrate over the contour. The problem involves […]