Series question

[jessie]

Determine whether the series converges or diverges. It it converges, find the sum.

Sum (n=1 to infinity) ln (1/n)

I know that you have to take the limit and there is a rule that states that ln(n)/n=0 but I don't quite get where to go from here

 

[pka]

Determine whether the series converges or diverges. It it converges, find the sum.

Sum (n=1 to infinity) ln (1/n)

I know that you have to take the limit and there is a rule that states that ln(n)/n=0 but I don't quite get where to go from here

\(\displaystyle \sum\limits_{n = 1}^\infty {\ln \left( {\frac{1}{n}} \right)} = \sum\limits_{n = 1}^\infty { - \ln \left( n \right)} = - \sum\limits_{n = 1}^\infty {\ln \left( n \right)}\)

So what do you think?

 

[jessie]

Does ln(n) diverge?

 

[pka]

Does ln(n) diverge?

Are you saying that you don't know?

If you are, the why in the world are you trying to do this level of question?

 

[jessie]

That was unnecessary, but I appreciate your help.

 

[Bob Brown MSEE]

Hi Jesse

Are you saying that you don't know?

If you are, the why in the world are you trying to do this level of question?

Welcome to FMH.
You are new, it will take some time for everyone to get to know your needs.
You would be surprised how many people just want their homework done for them.

Here's a hint.

n	ln(1/n)
1	0
2	-0.69
...	...

continue this table, notice that the sum of column #2 { called ln(1/n) } is the sum required. Do you see the trend?
Tell us what you observe.

 

[pka]

That was unnecessary

Quite to the contrary, it is absolutely necessary. You are clearly studying series convergence.
You ought have been told that \(\displaystyle [\sum\limits_{n = k}^\infty {{a_n}}\) converges only if \(\displaystyle (a_n)\to 0 \).
Now if you have not been taught that then you have been cheated and it was not by me.
This even has a name: the first test of divergence.
It is the very first test that is done in series convergence/divergence.

So you ought to know that \(\displaystyle (\ln(n))\to\infty \) thus \(\displaystyle \sum\limits_{n = k}^\infty {\ln (n)}\) must diverge.

So if you have not been taught this, then you have a case to complain.