Secretary Problems and High-order Sum of Harmonic Series
Background
The story
So I was reading this lecture note about secretary problems, and I was trying to prove Theorem 24.3 myself. I also obtained an sum representing the probability which is different from the one in the lecture note. However I had some difficulty proving that the two sums are equilvalent (if not given this problem). After a day of suffering, using the summation by parts technique, I found that the sum was in fact a high-order sum of the harmonic series, and that this sum has a pretty simple representation.