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August 30th, 2019, 11:45 AM  #1 
Member Joined: Aug 2016 From: Romania Posts: 32 Thanks: 1  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? 
August 30th, 2019, 12:17 PM  #2 
Senior Member Joined: Jun 2019 From: USA Posts: 387 Thanks: 212 
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. 
August 31st, 2019, 05:53 AM  #3 
Senior Member Joined: Dec 2015 From: Earth Posts: 834 Thanks: 113 Math Focus: Elementary Math 
$\displaystyle \mathop{\mathrm Σ}_{n=1}^\infty \frac{1}{n} > \int_{1}^{\infty} \frac{dx}{x}\rightarrow \infty$.


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fraction, positive, series, terms 
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