Is the Infinite Series Convergent or Divergent Despite Initial Undefined Terms?

[tsw303]

Homework Statement

Convergent/Divergent?

Problem 1: infinite series, sigma(n=1) [1/((n^2)-n)
This one is clearly convergent, however it is undefined at n=1.
Is the whole problem undefined? Or is it convergent from (n=2)?


Problem 2: infinite series, sigma(n=1) [ln(n)/n]
This is divergent. It works with the Limit Comparison Test, but for some reason, I'm thinking I have to start by using sigma(n=2) [ln(n+1)/(n+1)] because, otherwise the first term would be zero. Can the test for a positive series include zero?

Homework Equations

Problem 1: Sn=(1/0)+(1/3)+(1/8)+...
Sn - undefined

Problem 2: Sn=(0/1)+(ln2/2)+(ln3/3)+...
(positive series?)

The Attempt at a Solution

Problem 1: Sn undefined => Series also undefined?

Problem 2: LCT, use b=(1/n), lim(n..inf) (a/b) = lim(n..inf) [(lnn/n)/(1/n)] = lim(n..inf) ln(n) = inf , limit > 0 , sigma (1/n) divergent, so sigma (lnn/n) divergent too.
Series divergent

 

[foxjwill]

The convergance of a series only depends on what happens at the end.

 

[quantumdude]

Problem 1: infinite series, sigma(n=1) [1/((n^2)-n)
This one is clearly convergent, however it is undefined at n=1.
Is the whole problem undefined? Or is it convergent from (n=2)?

Yes, it's undefined. I think whoever wrote the problem botched this. You should ask your prof for clarification.

Problem 2: LCT, use b=(1/n), lim(n..inf) (a/b) = lim(n..inf) [(lnn/n)/(1/n)] = lim(n..inf) ln(n) = inf , limit > 0 , sigma (1/n) divergent, so sigma (lnn/n) divergent too.
Series divergent

The series does diverge, but you didn't prove it. In order for LCT to tell you anything, the limit has to come out to a finite, positive number. You won't get that here. Use the Integral Test instead.

 

[tsw303]

Thank you. Much appreciated.

What about the term "positive series"? The convergence tests call for positive series, but problem 2 has a_1 = 0. I can use a_n+1 for the proof, but do I have to?

 

[tsw303]

(...pardon me)

Is zero a positive number?

 

[quantumdude]

It really doesn't matter, because as foxjwill has pointed out, the convergence or divergence of a series only depends on what happens for large n. Deleting a finite number of terms from a series doesn't affect the convergence or divergence at all.

 

[quantumdude]

(...pardon me)

Is zero a positive number?

No, it isn't.

 

[rocomath]

(...pardon me)

Is zero a positive number?

It's neither positive or negative.