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This question already has an answer here:

I need some help to verify a question that I have done. It is to determine if a series ln(n)+5/n^2 is convergent or divergent. I have tried to use the limit comparison test. My syllabus has only touched on direct comparison,limit comparison test,integral test,ratio and root test as well as absolute and conditional convergence. Therefore, I can only use what's being taught to solve the question. Not really sure if my answer is correct. Would really appreciate it if someone could look through my solution to point out any mistake :) Question Solution

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marked as duplicate by GNUSupporter 8964民主女神 地下教會, lulu, Leucippus, Xander Henderson, positrón0802 Apr 1 '18 at 1:57

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Yes, your approach is correct.

You are applying the limit comparison test which is a reasonable test for your positive series.

You have also used L'Hospital Rule appropriately to get to your result.

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  • $\begingroup$ Thank you so much for the verification! I spent one hour to solve this question but I was very unsure of my answer and I would have felt really upset if I still got it wrong in the end. Thank you again for the reply! :) $\endgroup$ – Venerette Mar 31 '18 at 21:08
  • $\begingroup$ It was not an easy problem. Your time was well spent. $\endgroup$ – Mohammad Riazi-Kermani Mar 31 '18 at 21:12
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You solution seems correct indeed

$$\sum \frac{\ln(n)+5}{n^2}= \sum \frac{\ln(n)}{n^2}+\sum \frac{5}{n^2}$$

and

  • $\sum \frac{\ln(n)}{n^2}$ converges by limit compartison test with $\sum \frac{1}{n^{\frac32}}$

  • $\sum \frac{5}{n^2}$ converges

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