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August 30th, 2019, 10:45 AM   #1
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Two positive series with terms in a fraction

$$\mathop{\mathrm Σ}_{n=1}^\infty \frac{1+1/2+...+1/n}{n}$$
I have two series in a fraction and I do not understand how to solve this problem.I see that the numerator is a Harmonic series but that doesn't help me a lot.I tried doing the comparison test and I could only compare this series to:
$$\mathop{\mathrm Σ}_{n=1}^\infty \frac{1}{n}$$
and the result was the first series was bigger or equal than the harmonic series.
That means it diverges?How?Can someone explain?
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August 30th, 2019, 11:17 AM   #2
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It sounds like you've already answered your own question.

$\displaystyle \left( \frac{1}{n}+\frac{1}{2n}+\frac{1}{3n}+...+\frac{1} {n^2} \right) > \frac{1}{n}$
We know summing the RHS diverges (easy to find a proof online), so summing the LHS must diverge, as well. Anything "bigger than infinite" is also infinite.
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August 31st, 2019, 04:53 AM   #3
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$\displaystyle \mathop{\mathrm Σ}_{n=1}^\infty \frac{1}{n} > \int_{1}^{\infty} \frac{dx}{x}\rightarrow \infty$.
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