Notes on Homework #5
Exercise 2.7.1 c
We are going to prove the Alternating Series Test.
The hypotheses are that (
a
n
) is a
sequence with
a
n
≥
0,
a
1
≥
a
2
≥ · · ·
a
n
≥
a
n
+1
≥ · · ·
, and (
a
n
)
→
0.
1) The important thing to recognize is that the evennumbered partial sums end with a subtraction and
the oddnumbered ones end with an addition:
s
2
n
=
a
1

a
2
+
· · · 
a
2
n
,
s
2
n
+1
=
a
1

a
2
+
· · · 
a
2
n
+
a
2
n
+1
.
So
s
2
n
+2

s
2
n
=
a
2
n
+1

a
2
n
+2
,
which is nonnegative since
a
2
n
+1
≥
a
2
n
+2
. This shows that (
s
2
n
) is a monotone increasing sequence.
For the oddnumbered partial sums,
s
2
n
+1

s
2
n

1
=

a
2
n
+
a
2
n
+1
≤
0
because
a
2
n
+1
≤
a
2
n
, so (
s
2
n
+1
) is monotone decreasing.
Finally, since
s
2
n
+1
=
s
2
n
+
a
2
n
+1
and
a
2
n
+1
≥
0, we see that
s
2
n
≤
s
2
n
+1
. This verifies the inequalities in the hint:
s
2
≤
s
4
≤ · · · ≤
s
2
n
≤ · · · ≤
s
2
n
+1
≤ · · · ≤
s
3
≤
s
1
.
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 '08
 KBHANNSGEN
 Calculus, Algebraic Limit Theorem, s2n, Neven

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