Calculus.Determine whether the following series converges or diverges. Justify.

Question

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

Answer 1

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.

Answer 2

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