# Convergence of a Series

1. 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?

2. Homework Equations

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

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

3. 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
1. Homework Statement

2. Homework Equations

3. The Attempt at a Solution

Related Calculus and Beyond Homework Help News on Phys.org
The convergance of a series only depends on what happens at the end.

Tom Mattson
Staff Emeritus
Gold Member
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.

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?

(...pardon me)

Is zero a positive number?

Tom Mattson
Staff Emeritus
Gold Member
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.

Tom Mattson
Staff Emeritus