In the previous example,

we talked about two different sequences 
that occur inside an infinite series.

There's the sequence of individual terms

(those are the pieces that 
are being added together);

and then there's also the 
sequence of partial sums,

where S-n means 
the sum of the first n terms.

So S-4 would be the sum of
the first four terms and so on.

The idea is, if we look at the sequence 
of partial sums or the running total,

we can say, if the limit of that 
sequence (as n goes to infinity)

is equal to some number, 
which we call S, then the series

sum as n goes from 
one to infinity of ‘a’-n,

we say the series converges.

And we actually say that the value 
of that sum is this value right here.

We say that the series converges to S.

If the limit does not exist—

So if limit as n goes to infinity 
of S-n does not exist,

we say the series converges--
[corrects self] Sorry, diverges.