Convergence of a Series

In summary: However, for the purpose of convergence tests, a series with a first term of zero is considered a positive series. So in this case, it doesn't matter if we start at n=1 or n=2, the series will still be divergent. In summary, Problem 1 is an infinite series that is convergent, but undefined at n=1. It is unclear if the entire problem is undefined or convergent from n=2. Problem 2 is also an infinite series, but it is divergent. The Limit Comparison Test can be used, but the Integral Test is a better option. The first term being zero does not affect the convergence or divergence of a series.
  • #1
tsw303
7
0

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
 
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  • #2
The convergance of a series only depends on what happens at the end.
 
  • #3
tsw303 said:
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.
 
  • #4
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?
 
  • #5
(...pardon me)

Is zero a positive number?
 
  • #6
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.
 
  • #7
tsw303 said:
(...pardon me)

Is zero a positive number?

No, it isn't.
 
  • #8
tsw303 said:
(...pardon me)

Is zero a positive number?
It's neither positive or negative.
 

What is the definition of convergence of a series?

The convergence of a series is a mathematical concept that refers to the behavior of an infinite sequence of numbers. A series is said to converge if the terms of the sequence get closer and closer to a specific value, known as the limit, as the number of terms approaches infinity.

How can I determine if a series converges or diverges?

There are several tests that can be used to determine if a series converges or diverges, such as the comparison test, the ratio test, and the integral test. These tests involve analyzing the behavior of the terms in the series and can be used to determine if the series converges to a finite value or diverges to infinity.

What is the significance of the convergence of a series?

The convergence of a series is important in many areas of mathematics and science, as it allows us to make predictions and draw conclusions about the behavior of infinite sequences. In addition, many real-world phenomena can be modeled using infinite series, making the concept of convergence crucial in understanding and analyzing these phenomena.

What is the difference between absolute convergence and conditional convergence?

A series is said to have absolute convergence if the series of absolute values of its terms converges. This means that the series converges regardless of the order in which the terms are added. On the other hand, a series is said to have conditional convergence if it converges, but not absolutely. In this case, the series may converge or diverge depending on the order in which the terms are added.

Can a series converge to more than one value?

No, a series can only converge to one value. This value is known as the limit of the series. If a series converges to more than one value, it is said to be divergent. In other words, the limit of a convergent series is unique and cannot have multiple values.

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