Let and let denote the -th harmonic number. Examine the convergence of the series

**Solution**

We begin by the very well known

So is comparable to a constant multiple of . Thus, the series converges if that is to say and diverges otherwise.

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Let and let denote the -th harmonic number. Examine the convergence of the series

**Solution**

So is comparable to a constant multiple of . Thus, the series converges if that is to say and diverges otherwise.