Basic Analysis I — Real Analysis · Problem 5.4.6

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · euler-mascheroni-constant

Problem

a) Prove that for n∈Nn \in \mathbb{N}, ∑k=2n1k≤ln⁡(n)≤∑k=1n−11k.\sum_{k=2}^n \frac{1}{k} \leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}. b) Prove that the limit γ:=lim⁡n→∞((∑k=1n1k)−ln⁡(n))\gamma := \lim_{n\to\infty}\left(\left(\sum_{k=1}^n \frac{1}{k}\right) - \ln(n)\right) exists. This constant is known as the Euler--Mascheroni constant. It is not known if this constant is rational or not. It is approximately γ≈0.5772\gamma \approx 0.5772.

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