Consider the series
$$\displaystyle \sum_{n = 1}^\infty \exp \left (-\sum_{j = 1}^n \dfrac{1}{j} \right ) \dfrac{1}{n}$$
How do you determine whether this series is summable or not?
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What have you tried? Do you have any ideas of your own? Once we have a better idea of exactly where you're struggling, we can help you better ^_^ – HallaSurvivor Mar 19 '22 at 01:48
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1I have tried turning $\dfrac{1}{n}$ into $\exp(-\log(n))$ and combining with the exponential. I think I can use an upperbound by $\dfrac{1}{n^2}$ argument but I am not sure how to do this rigorously. – Coco Jambo Mar 19 '22 at 01:49
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1You should edit your question to include that attempt! That's very close to a full solution (it's exactly the right idea) and so it provides a lot of useful context that you have tried this problem yourself, as well as where exactly you need helping ^_^ – HallaSurvivor Mar 19 '22 at 01:50
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$\sum_{j = 1}^n \dfrac{1}{j}>\ln(n)$.
That gives you an upper bound for each term in your series.
Chris Sanders
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