Let denote the -th harmonic sum. Prove that

where denotes the Riemann zeta function.

**Solution**

Nothing more than a series manipulation. Successively we have:

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Let denote the -th harmonic sum. Prove that

where denotes the Riemann zeta function.

**Solution**

Nothing more than a series manipulation. Successively we have:

Recalling the fact that

we note that the series diverges.