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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. Thanks from idontknow August 31st, 2019, 05:53 AM #3 Senior Member   Joined: Dec 2015 From: Earth Posts: 834 Thanks: 113 Math Focus: Elementary Math $\displaystyle \mathop{\mathrm Σ}_{n=1}^\infty \frac{1}{n} > \int_{1}^{\infty} \frac{dx}{x}\rightarrow \infty$. Tags fraction, positive, series, terms Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Loren Number Theory 4 March 28th, 2017 03:09 PM leo255 Calculus 5 December 10th, 2014 02:55 AM 1love Algebra 8 May 20th, 2012 09:49 PM tomc Real Analysis 2 September 16th, 2009 06:36 PM dsjoka Calculus 1 May 16th, 2009 02:10 PM

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