Is there a formal definition of convergence of series?

One is often asked to check if a given series converges or not in one's first year of uni. Is there a formal definition that allows us to check this? We are only given a bunch of tests that are troublesome to remember (I've never liked cramming in math), and a Google-search only yields results regarding sequences.

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The formal definition is that the sequence of partial sums converges. This is usually stated in calculus but often the students don't really understand the concept of sequential convergence anyway and just end up memorizing a bunch of tests which they don't understand. This is why I have a big issue with the way series are taught in calculus. In many ways it's the most interesting part of the course but it ends up being the most painful part. By the way you are not alone, I had no idea what it really meant for a series to converge until I took real analysis. – Seth Apr 14 '14 at 19:12
A series is a sequence: the sequence of partial sums of another sequence. So the definition of convergence for series is the same as for sequences. – lhf Apr 14 '14 at 19:12
@lhf A series is a sequence? Not quite! And neither is their theory of convergence – imranfat Apr 14 '14 at 20:37

The formal definition is that $$\sum_{k=1}^\infty a_k$$ converges exactly if the sequence of it's partial sums converges, i.e. if the sequence $(s_n)_{n\in\mathbb{N}}$ defined by $$s_n = \sum_{k=1}^n a_k$$ converges. In other words, you per definition have that $$\sum_{k=1}^\infty a_k = \lim_{n\to\infty} s_n = \lim_{n\to\infty} \sum_{k=1}^n a_k$$

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Oh, of course, that was simple, yet something I was completely unaware of consciously. I have to agree with @Seth, it is indeed weird that we haven't formalized it. – Andrew Thompson Apr 14 '14 at 19:16
Isn't that tautological? As far as I know, the definition of $\sum_{k=1}^\infty a_k$ is $\lim_{n\to\infty} \sum_{k=1}^n a_k$. – Euro Micelli Apr 15 '14 at 0:33
@EuroMicelli Uh, yes, sure. But the OP asked for the definition of $\sum_{k=1}^\infty a_k$. I even emphasized that by writing specifically that "... you per definition have ...". – fgp Apr 15 '14 at 0:37
Ok. I read question as him looking for the definition of convergence of a series (in line with the title), not the definition of infinite series. I see your interpretation. – Euro Micelli Apr 15 '14 at 0:45
@EuroMicelli What I gave is the definition of convergence of a series. The definition of just an infinite series is that it's an element of $\mathbb{R}^\mathbb{N}$, i.e. a function from $\mathbb{N}$ into $\mathbb{R}$. – fgp Apr 15 '14 at 0:48

A series $a_n$converges if the limit $$\lim_{n\to \infty}\sum_{i=0}^n{a_i}=L$$ exists and is finite. In other words, for all $\epsilon>0$ there exists an $N$ such that for all $n>N$, $$\left|L-\sum_{i=0}^n{a_i}\right|<\epsilon$$

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Let $\sum_{n=m}^\infty a_n$ be a formal infinite series. For any integer $N\geqslant m$, we define the $N^{\text{th}}$ partial sum $S_N$ of this series to be $S_N:=\sum_{n=m}^N a_n$; of course, $S_N$ is a real number. If the sequence $(S_N)_{n=m}^\infty$ converges to some limit $L$ as $N\to\infty$, then we say that the infinite series $\sum_{n=m}^\infty a_n$ is convergent, and converges to $L$; we also write $L=\sum_{n=m}^\infty a_n$, and say that $L$ is the sum of the infinite series $\sum_{n=m}^\infty a_n$. If the partial sums $S_N$ diverge, then we say that the infinite series $\sum_{n=m}^\infty a_n$ is divergent.

from Terence Tao - Analysis I.

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