Question

Convergent or divergent?
A set of questions, my trials included

Answer 1

(1) Use divergence test: if \(\large \displaystyle \lim_{n \rightarrow \infty}u_n \neq 0\) then \(\large \displaystyle \sum_{n=1}^{\infty}u_n\)is divergent

Answer 2

Also use comparison test.
We know that the harmonic series \( \sum 1/n \) is divergent.
If all terms \( a_n \) of a given series is > 1/n, then the series is divergent.

Answer 3

I know those things, but I need detailed solutions or corrections to all of the questions

Answer 4

(1) evaluate \[\large \lim_{n \rightarrow \infty}\frac{ \ln n }{ \ln \ln n }\] using L'Hop's Rule

Answer 5

It's not a bad idea to apply those criteria to the problems, for example in question 1.
Try using the fact that
1/ln(n) > 1/n

Answer 6

You really should be asking about these questions one at a time.

Answer 7

I can't make 12 separate post

Answer 8

Exercises remain exercises. They are designed to help us learn by experience. Very often, having succeeded the first one or two, which are usually warm-ups, we will have the ability to do another two before more challenges.

So it is good to attack them one by one, and follow the learning experience, instead of treating them like a big pile of wood to be cut.

This is why wio suggests you to ask the questions one by one.

By the way, posting 12 at a time gives the impression that you are only interested in the answers, thereby not doing much for yourself, with the consequence of little interests in responses.

Finally, did you have time to attempt the first one with the comparison method that I suggested, since you already know all these techniques?