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Honestly, It is a homework problem.

Let $\sum_{n=1}^{\infty} x_n$ be a divergent series with positive terms. We have to examine whether the following are true or false

i) $\sum_{n=1}^{\infty} \frac{x_n}{1+n^2x_n}$ is convergent

ii) $\sum_{n=1}^{\infty} \frac{x_n}{1+nx_n} $ is divergent

I am able to do the first one by comparing with the series $\sum_{n=1}^{\infty} \frac{1}{n^2} $

But the second one I am unable to do.

If $x_n$ $\geq$ $1$ then of course it is divergent. But in general I am unable to do it.

Please help.

I really do not want the full steps.

I just need a hint.

(I have tried by Comparison test or its limiting form)

QTDA
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  • @Doug If I am not mistaken the inequality is obtained by assuming that $x_n >1$ which may not be so. Or am I missing something? – QTDA Jun 17 '22 at 16:00
  • You are correct. I deleted my comment.. Still struggling with it myself :) – Doug Jun 17 '22 at 16:03
  • Analyze cases when $nx_n\ge1$ or $nx_n<1$ – Ryszard Szwarc Jun 17 '22 at 16:31
  • Another hint: $x_n=1$ seldom and $x_n=2^{-n}$ otherwise. – Ryszard Szwarc Jun 17 '22 at 16:43
  • @Ryszard Szwarc I am still trying to analyse those hints. For $nx_n\geq1$ it can be compared to the series of $1/(2n)$ which is divergent. For $nx_n<1$, I am still trying – QTDA Jun 17 '22 at 16:48
  • Perhaps my first hint was misleading: split the indices into two subsets according to $nx_n\ge 1$ and $nx_n<1.$ You should be able to conclude that the terms of the series are located between ${1\over 2}\min(1/n,x_n)$ and $\min(1/n,x_n)$ – Ryszard Szwarc Jun 17 '22 at 17:00
  • @Ryszard Szwarc Okay. I got it but that is like a full solution. Hehe. Thank you so much for your help. I wish I could accept a comment as an answer. – QTDA Jun 17 '22 at 17:09
  • No problem. If you like a challenge show that if $x_n$ is decreasing and $\sum.x_n=\infty$ than the series $\sum \min(x_n,1/n)$ is always divergent unlike in the case when $x_n$ does need to be decreasing. – Ryszard Szwarc Jun 17 '22 at 17:24
  • Okay I will try. – QTDA Jun 17 '22 at 17:33
  • @Koro yes it does. Thank you – QTDA Jun 18 '22 at 05:06

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