What does it mean for a series to be convergent?

I have the definition:

Let $(a_n)$ be a sequence of real numbers. Let $s_n=a_1+a_2+...+a_n$. We say the series $a_1+a_2+...$ is convergent if the sequence of partial sums $(s_n)$ is convergent. The limit of this sequence is called the sum of series.

What does this mean in terms of applications to a series? How does one compute the limit of this sequence from the definition?

• The definition isn't used to find limits, it is used to prove that one's guess at a limit actually works. – Git Gud May 27 '14 at 13:39
• @GitGud Okay, I see. How does one find a limit for a convergent series? – user127700 May 27 '14 at 13:41
• There is no general method,@user127700: some kinds of series one can do this, some others one can do that...and in many,many cases nobody has the slightest idea. – DonAntonio May 27 '14 at 13:42
• It is a very difficult task, in general. For example, it is well known that $\sum (1/n^3)$ converges, but there is no known closed espression for the limit. You can, of course, approximate it computng the sum of "many" terms. The more approximation you need, the more terms you have to sum. – ajotatxe May 27 '14 at 13:45
• @DonAntonio I am aware of the Cauchy test and the Ratio test. I wanted to be sure I could say as much as possible about convergence, rather than just stating whether or not they converge. Your answer has helped greatly. – user127700 May 27 '14 at 13:47

The statement that the partial sums $(s_n)$ are convergent is a statement that they form a Cauchy sequence. This means for any $\epsilon > 0$ there is an $N \in \mathbb{N}$ for which $$|s_n - s_m| < \epsilon$$ for all $n > m > N$. In particular if we rewrite $$s_n - s_m = \sum_{k=1}^n a_k - \sum_{k=1}^m a_k = a_{m+1} + a_{m+2} + \cdots + a_n = \sum_{k=m+1}^n a_k$$ Then we say a series converges if for all $\epsilon > 0$ there is an $N$ for which $$\left|\sum_{k=m+1}^n a_k\right| < \epsilon$$ for all $n>m>N$.
• In particular, since $n$ can be as large as we want (much larger than $m$) we can think heuristically that a series converges provided its tails get arbitrarily small. – Joel May 27 '14 at 13:48
• So by letting n and m tend to infinity, we can see that $s_n - s_m = s - s = 0.$ (Where $s = lim s_n = lim s_m)$ Is that correct? – user127700 May 27 '14 at 13:51