Well, first question is: convergence of what? Do we talk about a convergent sequence of "numbers", which can be vectors as well, or do we talk about sequences of functions. All we need is a sort of measurement, that allows us to say whether two objects are closer to each other than other objects.
You might want to read these Wikipedia entries on it
- https://en.wikipedia.org/wiki/Sequence#Limits_and_convergence
- the links in https://en.wikipedia.org/wiki/Function_series
As you can see there, it is a bit messy. Basically it is about a limit and its definition:
"##a## is the limit of a sequence ##(a_n)_{n \in \mathbb{N}}##" is written ##\lim_{n \rightarrow \infty} a_n = a## and means
$$ \forall \, \varepsilon > 0 \, \exists \, N(\varepsilon) \in \mathbb{N} \,\forall \, n \geq N(\varepsilon) : \,\vert \,a_n - a \,\vert \, < \varepsilon$$
which means, for any given margin of accuracy ##\varepsilon## there can be found a natural number ##N(\varepsilon)## from which on all elements of the sequence, i.e. those that come after, are all within the given distance ##\varepsilon## to the limit point ##a##. For short: the higher the ##n## the closer the ##a_n## are to ##a## or more precisely: the closer you want to get to the limit, the higher the ##N## is from where on you're close enough. But there always is one.
You see, all it takes is ##\,\vert \, . \, \vert## to measure a distance. This includes vectors, functions, series and so on. All of them are some kind of generalization to this concept, or variations of how the convergence behaves (as in the case of functions: it could be the case, that a sequence ##f(x_n) \rightarrow f(x)## converges faster than a sequence ##f(y_n) \rightarrow f(y)## for the same function at different points.)
But whether the elements of ##(a_n)_{n \in \mathbb{N}}## are numbers, vectors or function values of which dimension ever, or partial sums ##(\Sigma_{k = 1}^n a_k)_{n \in \mathbb{N}}## doesn't matter.