The sequence $H_n-\ln(n)$ converges

Question

Is there a proof that the sequence $\displaystyle \sum_{k=1}^n \frac{1}{k}-\ln(n)$ converges that doesn't use integrals?

Answer 1

Since $\ln(n+1)-\ln n \to 0$, we can just as well put $\ln(n+1)$ there. Write the sum as $$ \sum_{k=1}^n\left( \frac{1}{k} - \ln \frac{k+1}{k }\right)$$ By Taylor's formula applied to $f(x)=\ln(1+x)$, $$\ln\left(1+\frac{1}{k}\right)=\frac{1}{k} + \frac{1}{2k^2} f''(\xi) $$ where $\xi$ is between $0$ and $1/k$. Since $|f''(\xi)|\le 1$, it follows that $$\left|\frac{1}{k} - \ln \frac{k+1}{k }\right| \le \frac{1}{2k^2}$$ which by Comparison test implies that the series $$ \sum_{k=1}^\infty\left( \frac{1}{k} - \ln \frac{k+1}{k }\right)$$ converges.