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Limit with harmonics

Let \mathcal{H}_n denote the n-th harmonic number. Evaluate the limit

    \[\ell = \lim_{n \rightarrow +\infty} \left ( \mathcal{H}_n - \frac{1}{n} \sum_{k=1}^{n} \mathcal{H}_k \right )\]

Solution

It follows from Cesaro and the definition of the Euler Mascheroni constant that the RHS is equal to

    \[\left ( \mathcal{H}_n -\ln n \right ) - \frac{1}{n} \sum_{k=1}^{n} \left ( \mathcal{H}_k -\ln k \right )+ \ln n - \frac{1}{n} \ln n!\]

The first two terms tend to \gamma - \gamma =0. All that is left to evaluate the limit of \ln n - \frac{\ln n!}{n} which equals 1 from here.

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