# How to prove $x_n$ converges as $n \to \infty$. [duplicate]

Let $x_n = \sum_{k=1}^{n} \frac{1}{k} - \log n$. Prove that $x_n$ converges as $n \to\infty$.

• The (finite) limit $\gamma$ of the sequence $\{x_n\}$ is well-known: it is the Euler constant. Check here: en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant – Ángel Valencia Dec 1 '15 at 1:55
• Do you know Stieltjes integration? – Clayton Dec 1 '15 at 1:55
• @ÁngelValencia So if the limit is well known, why would this be false? – 5xum Dec 1 '15 at 1:56
• Sorry, it was my mistake. I didn't read well. – Ángel Valencia Dec 1 '15 at 1:56
• Try using left and right Riemann sums to bound $\sum_{k=1}^n \frac{1}{k}$ by something involving $\log (n)$ (since $\int_1^n \frac{1}{x}dx= \log(n)$). – kccu Dec 1 '15 at 2:01

• Show that $$\frac{1}{n + 1} < \log\left(1 + \frac{1}{n}\right) < \frac{1}{n}. \tag{1}$$ You might need the facts that the sequence $\left(1 + \frac{1}{n}\right)^n$ increasingly converges to $e$ and the sequence $\left(1 + \frac{1}{n}\right)^{n + 1}$ decreasingly converges to $e$.
• Use $(1)$, show that $$\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n + 1} < \log(n + 1) < 1 + \frac{1}{2} + \cdots + \frac{1}{n}. \tag{2}$$
• Use $(2)$, show that the sequence $\{x_n'\}$ is increasing and bounded above, where $$x_n' = 1 + \frac{1}{2} + \cdots + \frac{1}{n} - \log (n + 1).$$ Hence $x_n'$ converges.
• Show that $x_n$ and $x_n'$ have the same limit.
• Good outline. An alternative route is to show that $x_n' = \int_{1}^{n+1}\left[\frac{1}{\lfloor x\rfloor} - \frac{1}{x}\right]{\rm d}x$ and bound the integrand above to conclude that $x_n'$ converges. – Kibble Dec 1 '15 at 2:25