Consider the sequence $\{a_n\}$ and the sum of its first $n$ terms, \[ s_n = \sum_{k=1}^n a_k=a_1+\cdots+a_n. \] The sequence $\{s_n\}$ is called an infinite series with $n\text{th}$ term $a_n$ and partial sum $s_n.$

For example, given $a_n=n,$ and $$s_n=\sum_{k=1}^n k=1+2+3+\cdots+n,$$ then $\{s_n\}$ is an infinite series with $n\text{th}$ term $a_n=n$ and $n\text{th}$ partial sum $s_n.$

Rather then refer to a series in terms of $s_n$, it is customary to refer it in terms of $a_n,$ and we write $\sum a_n$ for the series $\{s_n\}.$

Thus, $\sum n$ is the series from the example since $a_n=n$. The $n\text{th}$ partial sum of this series is just the sum of the first $n$ positive integers, a famous formula of which is $$s_n = \sum_{k=1}^n k=\frac {n(n+1)}2$$ This formula is given without proof, but can be verified by mathematical induction. Notice that $\{s_n\}$ is the sequence $$1, 3, 6, 10, 15, 21, \ldots$$ in contrast to $\{a_n\}$ which is $$1, 2, 3, 4, 5, \ldots$$

When talking about sequences and series, it is important to distinguish between a sequence $\{x_n\},$ it's related series $\sum x_n$, and the $n\text{th}$ term of both, $x_n.$ For example, $$\{a_n\}\ne a_n$$ and $$\sum a_n = \{s_n\}\ne s_n=\sum^n a_k.$$

Note the shorthand $\{x_n\}=\{x_n\}_{n=1}^\infty$ and $\sum x_n=\sum_{n=1}^\infty x_n.$ Furthermore, while $s_n$ is the $n\text{th}$ term of the sequence $\{s_n\},$ we say that $a_n$ is the $n\text{th}$ term of the series $\{s_n\},$ i.e. of the series $\sum a_n.$

For instance, given the sequence $\{n\}$ of positive integers, we may speak of the series $\sum n$ by which we roughly mean an "infinite sum" of those same positive integers, $$1 + 2 + 3 + \cdots + n + \cdots$$

Now, we may ask, does the series $\sum n$ converge? By this we mean, does the infinite process of adding "all" the positive integers, of which there are an infinite number, lead to some finite sum, a number?

Intuitively, each time we add the next positive integer to the sum before it, that next sum is larger, in turn, than the one before it. Thus, we see that the sequence of partial sums $s_n$ $$1, 3, 6, 10, 15, 21, \ldots$$ grows without bound. Intuitively, then, the series $\sum n = 1 + 2 + 3 + \cdots + n + \cdots$ does not converge, and we write $$\sum n = \infty.$$

In general, if a series $\sum a_n$ does not converge, we say it diverges and write $$\sum a_n=\infty.$$

Notice how, in the example where $a_n = n,$ it was necessary to reason about the behavior of the sequence of partial sums $\{s_n\}$ to determine whether the series $\sum a_n$ converged. This is typical. In practice, the sequence of partial sums $\{s_n\}$ is the key to determining whether the series $\sum a_n$ converges.

Specifically, we say that the series $\sum a_n$ converges if it's sequence of partial sums, $\{s_n\}$ does.

Therefore, to understand what it means for a series to converge, what "infinite addition" means or an "infinite sum", we must define what it means for a sequence to converge, because convergence of $\sum a_n$ is defined in terms of convergence of the sequence $\{s_n\},$ its sequence of partial sums.

It's actually not very hard to get an intuitive notion of what it means for a sequence to converge. For instance, consider the sequence \[ \left\{ \frac 1 n\right\}=1, \frac 1 2, \frac 1 3, \frac 1 4, \ldots, \frac 1 n, \ldots. \]

Intuitively, this sequence seems to converge to 0, because it is never greater than 1 and never less than 0, and it is always decreasing. We say the sequence is bounded because it has both upper and lower bounds. We say it is monotonically decreasing since it never starts increasing after decreasing.

So we might conjecture that a bounded monotonically decreasing sequence converges to its lower bound. Similarly, we might conjecture that a bounded monotonically increasing sequence converges to its upper bound. Both statements are true.

Thus, we say the series $\sum a_n$ with $n\text{th}$ partial sum $\frac 1 n,$ converges to $0,$ and write $$\sum a_n =0$$ to indicate this.

This does not say that the different series $\sum s_n$ converges, which happens to be false when $s_n = \frac 1 n.$

That is to say, the series $\sum\frac 1 n$ does not converge.

Thus, although we saw that, intuitively, the sequence $\left\{s_n=\frac 1 n\right\}$ converges to $0$, we did not claim the series $\sum s_n$ converges. In fact it does not, so we write $$\sum \frac 1 n=\infty.$$

Instead, the claim was that, since $\left\{s_n = \frac 1 n\right\}$ converges to 0, then the series $\sum a_n$ whose $n\text{th}$ partial sum is $s_n$ does as well, and in particular it converges to 0: $$\sum a_n=0$$

We have not given a formula for $a_n$ in this particular example because it's irrelevant to the point. Suffice it to say that $a_n \ne s_n = \frac 1 n$ since $\sum a_n=0$ but $\sum s_n = \infty.$

Having now given examples of both convergent and divergent sequences and series, we conclude.