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.