Example
Consider the following two infinite series:
(a)
∞
X
n
=0
2
9
n
(b)
∞
X
n
=1
3
n

1
(

7)
n
.
For each series, find its sum and tell whether it converges or diverges.
Solution:
Each of the series in
(a)
and
(b)
is a geometric series. Recall that a geometric series may be
written as
∞
X
n
=0
ar
n
or
∞
X
n
=1
ar
n

1
(the two summations are equivalent), and that if

r

<
1, the series converges with sum given
by
∞
X
n
=0
ar
n
=
∞
X
n
=1
ar
n

1
=
a
1

r
.
Otherwise, if

r
 ≥
1, the geometric series diverges.
(a)
∞
X
n
=0
2
9
n
Because the summation for this series begins with the
n
= 0 term, it is natural to try to
express it in the form
∞
X
n
=0
ar
n
. Since
X
n
=0
2
9
n
=
∞
X
n
=0
2
·
1
9
n
,
it must be that
a
= 2
,
r
=
1
9
,
so that

r

<
1. Thus the series converges and its sum is given by
∞
X
n
=0
2
·
1
9
n
=
2
1

1
9
=
2
8
9
=
18
8
=
9
4
.
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 Fall '10
 KIHYUNHYUN
 Calculus, Geometric Series, Summation, n=1

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