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Convergence of a Series

  • Thread starter tsw303
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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
 

Answers and Replies

354
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The convergance of a series only depends on what happens at the end.
 
Tom Mattson
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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.
 
7
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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?
 
7
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(...pardon me)

Is zero a positive number?
 
Tom Mattson
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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
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