how to show that a series converges without use of limits.

Just wanted to know if there is another method?

Of the methods I know:

1. Ratio test
2. Comparison test
3. Root test
4. Integral test
5. Limit comparison test

All make use of limits.

The reason why I am asking question, is because my class mate told the teacher that limits is not on the syllabus thus he should spend time talking about it, and focus on the test. (a level)

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I'd say series convergence inherently involves limits... – Jakub Konieczny Sep 10 '12 at 7:09
When we say that a sequence converges, we mean that it converges to a limit, so it is rather difficult to discuss convergence in any generality without. If you are working over $\mathbb R$ and have a sequence is increasing and you can show that it is bounded above, then you know it has a limit (similarly decreasing and bounded below). – Mark Bennet Sep 10 '12 at 7:09
There is a criterion that says that if you have a decreasing sequence of positive $(a_n)_{n \in \mathbb{N}}$ reals, then the series $\sum_{n = 0}^\infty (-1)^n a_n$ converges, provided that $\lim a_n = 0$ (which is usually straightforward to check). Is that the type of result you mean? – Jakub Konieczny Sep 10 '12 at 7:12
Wikipedia is your friend, as always: en.wikipedia.org/wiki/… – Jakub Konieczny Sep 10 '12 at 7:19
@MaoYiyi Well if you are working in a complete metric space and can show your series is cauchy then you know it converges to something. – user38268 Sep 10 '12 at 9:01