One fine summer morning not so long ago, I woke up with a start to the stark realization that I had no idea how to derive the formula for the sum of powers of natural numbers. I am talking about a closed-form expression for the series (for non-negative integer \(p\)) \[ \sum_{i=1}^n {i^p} = 1 + {2^p} + {3^p} + \dots + {n^p} \]

Years back, I encountered the following problem in a book on Number Theory: Prove that the sum \(H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\) is never an integer after \(n > 1\). In other words, in the sequence \(1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \dots\), there is only one integer, and that is the first number \(1\).