Are there any series that do not diverge or converge?

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In summary, the conversation discusses the definition of convergence and divergence as it applies to series. The participants also mention the difference between divergence that goes off to infinity and divergence where the partial sums do not settle down. They also mention the property of an always increasing partial sum leading to convergence, and the possibility of a series diverging to a finite sum or not converging to anything. The conversation also touches on the concept of countable and uncountable infinity in relation to series.
  • #1
flyingpig
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Homework Statement



Keep this at a Calc II level, I thought about this when I was on Mathematica, because it seems it can only give boolean answers with SumConvergence. So are there series which do not diverge or converge?
 
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  • #2
What is the definition of 'diverge' as it applies to a series?
 
  • #3
Most math texts will give us a nice definition for a convergent series involving the convergence of the sequence of partial sums. They will then follow that definition with a line something like this: "A series that does not converge is said to diverge." If we use this definition of convergence and divergence, then there can not be anything that does not either converge or diverge.
 
  • #4
I think diverge means having no finite sum
 
  • #5
flyingpig said:
I think diverge means having no finite sum
Look at what WolframAlpha and Wikipedia say about 'divergent series'. You'll find that their definitions agree with what kru_ stated.
 
  • #6
what about an alternating series involving sine or cosine. If I took the improper integral of sin(x) or cos(x) with infinity in one of the bounds. It wouldn't converge or diverge.
 
  • #7
First off, I'm assuming the context is infinite series, not sequences.

To diverge, it doesn't have to go off to [itex]\pm oo[/itex].
The sum 1 -1 +1 -1 +1 ... diverges.
So informally, one might say we see two kinds of divergence. Divergence which goes off to infinity
and divergence where the partial sums don't settle down.

Related:
In higher math, we see a property, if the partial sum is always increasing (all terms are positive) and the series or sum has an upper bound, then the series converges.
 
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  • #8
nickalh said:
First off, I'm assuming the context is infinite series, not sequences.

To diverge, it doesn't have to go off to [itex]\pm oo[/itex].
The sum 1 -1 +1 -1 +1 ... diverges.
So informally, one might say we see two kinds of divergence. Divergence which goes off to infinity
and divergence where the partial sums don't settle down.

Related:
In higher math, we see a property, if the partial sum is always increasing (all terms are positive) and the series or sum has an upper bound, then the series converges.
If it diverges what does it diverge too. Why couldn't I just say it converges to 0?
 
  • #9
It doesn't have to diverge to anything.
 
  • #10
so we are saying that if it doesn't converge it diverges . Is this a definition from calculus?
And when we take it out to infinity are we taking it to countable infinity or uncountable infinity, I am just wondering.
 
  • #11
This is the definition for a case of a series diverging, but it's similar to the cases of limits, sequences, and improper integrals. See http://www.mathwords.com/d/diverge.htm"
 
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  • #12
1=1
1-1 =0
1-1+1 =1
1-1+1-1=0
1-1+1-1+1=1
...
Does that series appear to be converging to zero or going to zero?



And when we take it out to infinity are we taking it to countable infinity or uncountable infinity, I'm just wondering.
I'm assuming by take it out to oo, you mean count up all the terms, and not what is the sum.
Are there a countable number of terms or uncountable number of terms? In this context, I interpret countable to mean, it is possible to put a unique integer index on each term.
 
  • #13
ok so it would be countable infinity .
 

FAQ: Are there any series that do not diverge or converge?

What does it mean for a series to diverge or converge?

When we say a series diverges, it means that the sum of its terms grows larger and larger without bound. In other words, the series does not have a finite sum. On the other hand, a series converges when the sum of its terms approaches a finite value as we take more and more terms into account.

How can we 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 comparing the given series to a known convergent or divergent series or using calculus techniques to evaluate the series.

Are there any series that do not diverge or converge?

Yes, there are some series that do not diverge or converge, but instead oscillate between different values. These are known as oscillating or alternating series. An example of this is the alternating harmonic series, which has a sum of ln(2).

Can a series converge in one situation and diverge in another?

Yes, a series can converge in one situation and diverge in another. This is because the convergence or divergence of a series depends on the values of its terms and the specific conditions or constraints given.

Why is it important to determine if a series converges or diverges?

Determining if a series converges or diverges is crucial in many applications, especially in science and engineering. For example, in physics and engineering, series are used to approximate real-life phenomena, and knowing if the series converges or diverges helps us determine the accuracy of these approximations. In mathematics, the convergence or divergence of a series is also essential in understanding the behavior of infinite sums and in proving theorems.

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