
17-16 Calculus – Differentiation and Integration
17.4.4 Geometric Series
Any series of the form ar
n
n=
∞
∑
0
or ar
n
n
−
=
∞
∑
1
1
is called geometric series.
Example 17.18 Show that the geometric series converges for r <1 and diverges for r ³ 1.
Solution: A geometric series can be denoted as ar
n
n=
∞
∑
0
.
Assume that r <1, i.e., -1 < r < 1.
We have
s ar a r r r r
a r r r r
r
r
a
r
n
i
i
n
n
n
=
∑
= + + + + +
= + + + + +
−
−
=
−
=1
2 3
2 3
1
1
1
1
1
( )
( )
(( )
( )
1
1
1
2 3 2 3 1
1
+ + + + + − − − − − −
=
−
−
=
−
−
+
+
+
r r r r r r r r r
a
r
r
a
ar
n n n
n
n
11
−
Now, lim lim lim lim .
n
n
n n
n
n
n
s
a
ar
a
ar
r
→∞ →∞ →∞
+
→∞
=
−
−
−
=
−
−
−
1
We have by known theorem that {r
n
} converges to 0 for -1 < r < 1. ...