
17-26 Calculus – Differentiation and Integration
17.5.6 Alternating Series Test
If Su
n
is a series of positive terms such that u
n
³ u
n+1
for all n and lim
n
n
u
→∞
= 0 , then the series
( )−
∑
+
1
1n
n
u converges.
Example 17.37 Test the series
( )−
∑
+
=
∞
1
1
1
n
n
n
for convergence.
Solution: Let
u
n
n
=
1
for all n.
Also u
n
³ u
n+1
for all n. And
lim lim
n
n
n
u
n
→∞ →∞
= =
1
0
Therefore, by alternating series test
( )
( )
−
∑
=
−
∑
+
=
∞
+
=
∞
1
1
1
1
1
1
n
n
n
n
n
u
n
is convergent.
Example 17.38 Test the series
cos( )n
n
n
p
=
∞
∑
2
for convergence.
Solution We know
cos( ) ( ) .n
n
p = −1
Thus, let u
n
n
=
1
for all n.
Also u
n
³ u
n+1
for all n. And
lim lim
n
n
n
u
n
→∞ →∞
= =
1
0
Therefore, by alternating ...