Series: A Series (Or Infinite Series) is
obtained from a sequence by adding the terms of the sequence. Another
sequence associated with the series is the sequence of partial
sums. A series converges if its sequence of partial sums
converges.
The sum of the series is the limit of the sequence of partial sums.
For example, consider the geometric series defined by the sequence
Then the -th
partial sum is given by and, for ,
.
Thus
and
.
Another situation in which we we can actually compute the partial
sums occurs if those sums are collapsing. It may not be obvious
that that is the case, but look for it in this example:
=
and thus
=
.