# Homework Help: Are there any series that do not diverge or converge?

1. Jul 18, 2011

### flyingpig

1. The problem statement, all variables and given/known data

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?

2. Jul 18, 2011

### SammyS

Staff Emeritus
What is the definition of 'diverge' as it applies to a series?

3. Jul 18, 2011

### kru_

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. Jul 19, 2011

### flyingpig

I think diverge means having no finite sum

5. Jul 19, 2011

### SammyS

Staff Emeritus
Look at what WolframAlpha and Wikipedia say about 'divergent series'. You'll find that their definitions agree with what kru_ stated.

6. Jul 23, 2011

### cragar

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. Jul 23, 2011

### nickalh

First off, I'm assuming the context is infinite series, not sequences.

To diverge, it doesn't have to go off to $\pm oo$.
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.

Last edited: Jul 23, 2011
8. Jul 23, 2011

### cragar

If it diverges what does it diverge too. Why couldn't I just say it converges to 0?

9. Jul 23, 2011

### SammyS

Staff Emeritus
It doesn't have to diverge to anything.

10. Jul 23, 2011

### cragar

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, im just wondering.

11. Jul 23, 2011

### SammyS

Staff Emeritus
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" [Broken]

Last edited by a moderator: May 5, 2017
12. Jul 23, 2011

### nickalh

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?

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. Jul 24, 2011

### cragar

ok so it would be countable infinity .