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A Caution on the Alternating Series Test
Theorem 14 (The Alternating Series Test) of the textbook says:
The series
∞
X
n
=1
(

1)
n
+1
u
n
=
u
1

u
2
+
u
3

u
4
+
···
converges if
all
of the following conditions are satisﬁed:
1.
u
n
>
0
for all
n
∈
N
.
2.
u
n
≥
u
n
+1
for all
n
≥
N
, for some integer
N
.
3.
u
n
→
0
as
n
→ ∞
.
Clearly, to show the convergence, you need to check all these three conditions.
What can you conclude if a given alternating series fails one or more of the
three conditions? Is that series divergent? Unfortunately, the failure of this
test does not immediately lead to the divergence! You need to use some other
test, often, the
n
th Term Test for Divergence
to conclude that the series
diverges. The reason is the following. The equivalent statement to Theorem
14 is that:
(1)
“If a given alternating series diverges, then it fails to satisfy one or
more of the above three conditions.”
This is diﬀerent from the following statement, which is
false
:
(2)
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 Spring '08
 Saito
 Calculus

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