Comparision test for convergence

I was wondering whether one showing series is greater than another is sufficent to show if the former converges, so does the latter. Thats what I took the proof in the link to imply. Cauchy condensation test - Wikipedia, the free encyclopedia. But I've figured that must be wrong. So how does this proof justify convergence simply by noting one series is bigger than the other?

Thanks

What do you mean exactly by "a series is greater than another". It works if the are nonnegative numbers.

If for all n, ("for all but a finite n" is sufficient) and , then if converges, so does .

I mean, as it shows on the link, the infinite sum. The difficulty I have with the proof is that it doesn't compare to , it just refers to the size of the series.

So if you show that the is greater than or equal to , does this imply for ?

Quote:

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Originally Posted by Duke

So if you show that the is greater than or equal to , does this imply for ?

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No it does not. That is backwards.
. Not the other way round.

ok thanks for that. I am still confused about 2 things. One is that does not always exist so how does the implication work then. Second, the proof of the cauchy condensation theorem (on wikipedia) does not compare each term in with the corresponding term in so how does the proof work?