An infinite series is the sum of the terms of an infinite sequence,
We determine whether it converges or not by adding up finitely many terms in what is known as a partial sum. We can add 1, 2, 3, ..., and n terms, and so on. This process generates a sequence of partial sums, and hence we can talk about the convergence of that sequence. If the sequence of partial sums has a limit, then the infinite series is convergent.
Given a series 
. Let 
denote its 
partial sum:
If the sequence 
is convergent and 
is a
real number, then the series 
is convergent, and we
write
The number s is called the sum of the series. Otherwise, the series is called divergent.
One famous series is the geometric series, which is defined as
Another famous series is the harmonic series, which is defined as
The geometric series
is convergent if |r|<1, with sum
It is divergent otherwise.
If the series 
is convergent, then
. By contrast then, if
fails to exist, or exists but is different from 0,
then the series diverges. Unfortunately, however,
is not enough to guarantee that the series
converges. An example is the harmonic series.
If 
and 
are convergent series and c is a
constant, then so are
One method for showing that a series is divergent is by comparison with some known divergent series (e.g. the harmonic series).
Series are important, but exceedingly odd. What does it mean to add up an infinite number of things? It is a bizarre notion, which troubled Greeks like Zeno, who used the confusion to prove that motion is possible and impossible, that a tortoise can always beat a hare in a race, etc.
We think that we've got a handle on it now, thanks to the ideas of partial sums, and convergence of sequences. I hope that we do!