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Prove that the sequence converges.

For each positive integer $n$, let $$y_n = 1 + \frac12 + \frac13 + \cdots + \frac1n - \int_1^n \frac{dx}x.$$

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    Did you attempt the question? Can you show us your work and where you're stuck? HINT: One approach for this problem would be to prove that the sequence is monotone (decreasing) and bounded.2011-12-09
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    [Very related...](http://math.stackexchange.com/questions/55358)2011-12-09
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    Show the sequence is monotone decreasing. Since the sequence in monotone decreasing, it is Riemann integrable. Since the sequence is Riemann integrable, it is bounded. Since the sequence is bounded and monotone, it is convergent.2011-12-09
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    [This](http://math.stackexchange.com/questions/56688) is also related.2011-12-09

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