Are there any series that do not diverge or converge?

[flyingpig]

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?

 

[SammyS]

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

 

[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.

 

[flyingpig]

I think diverge means having no finite sum

 

[SammyS]

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.

 

[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.

 

[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.

 

[cragar]

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.

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

 

[SammyS]

It doesn't have to diverge to anything.

 

[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, I am just wondering.

 

[SammyS]

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"

 

[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?

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.

 

[cragar]

ok so it would be countable infinity .