I'm trying to calculate the complexity of an algorithm. I've managed to boil down the cost of the algorithm to the following sum.
$\sum_{k = 1}^{\log_{2}(n)} \frac{n\log_{2}(\frac{k^{2}}{2})}{k^{2}}$ where $n$ is the size of the data set.
I'm struggling to turn the sum into a function that I can use to figure out the big-$O$ of my algorithm. I was initially thinking that if I could convert the sum into a geometric series, then I might have something that I could work with.