A series converges if its sequence of partial sums con-
verges.
The sum of the series is the limit of the sequence of
partial sums.
For example, consider the geometric series defined by the se-
quence
a
n
=
1
r
n
,
n
=
0
,
1
,
2
,...
Thus
∑
∞
k
=
0
(
2
π
)
k
=
and
∑
∞
k
=
2
(
2
π
)
k
=
.
Another situation in which we we can actually compute the par-
tial sums occurs if those sums are
collapsing
. It may not be
obvious that that is the case, but look for it in this example:
∑
n
k
=
0
1
k
2
+
9
k
+
20
=
and thus
∑
∞
k
=
0
1
k
2
+
9
k
+
20
=
.

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