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2020

Partial Summation and a Probabilistic Example

6 minute read

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A very useful result used often in analytic number theory is the so-called partial summation formula. It is a technique used to change sums into integrals, so that we can then use the nice results of calculus to attack the integral more effectively. We know that \begin{equation*} \sum_{n=1}^N 1 = N, \qquad \sum_{n=1}^N n = \frac{N(N+1)}2, \end{equation*} and we have similar results for $\sum_{k\leqslant N} n^k$ where $k$ is a positive integer, but can we get a nice formula for sums like \begin{equation*} \sum_{n=1}^N \frac 1n \qquad \text{or} \qquad \sum_{n=1}^N \log n? \end{equation*} What about \begin{equation*} \sum_{n=1}^N \sqrt{n}? \end{equation*} This is what partial summation will help us achieve. … Read more