For each positive integer n, let $\gamma_n = 1+\frac12+ \cdots + \frac1n - \int_1^n \frac1x \, dx$. Prove that the sequence $\{\gamma_n\}$ converges.


marked as duplicate by Omnomnomnom, Trevor Wilson, Daniel Fischer, gt6989b, Namaste Nov 18 '13 at 19:01

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    $\begingroup$ I suspect this is a duplicate question. At any rate, phrasing it in this imperative way or other ways suitable primarily for assigning homework tends to be frowned on here. Giving your own thoughts on the matter will make a better impression on the regulars. $\endgroup$ – Michael Hardy Nov 18 '13 at 18:34

Here's a hint to a possible approach. Consider the sequence $\{a_{n}\}$ with $a_{n}=\left\{\begin{array}{ll} \frac{1}{n} & : n \text{ odd}\\ \int_{n-1}^{n}\frac{1}{x}dx & : n \text{ even}\end{array} \right.$

Can you show that $\{a_{n}\}$ is decreasing?


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