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Analysis 1: Exercises 14
1.
Let (
a
(
n
)) be a decreasing sequence of strictly positive numbers with
n
th partial sum
s
(
n
).
(a) By grouping terms in two diﬀerent ways, show that
1
2
{
a
(1) + 2
a
(2) +
···
+ 2
n
a
(2
n
)
}
6
s
(2
n
)
6
±
a
(1) + 2
a
(2) +
···
+ 2
n

1
a
(2
n

1
)
²
+
a
(2
n
)
.
(b) Use (a) to show that
∞
X
n
=1
a
(
n
) converges if and only if
∞
X
n
=1
2
n
a
(2
n
) converges.
This result is called the
Cauchy condensation test
.
2.
Let
s
be the sum of the alternating series
∑
∞
n
=1
(

1)
n
+1
a
n
with
n
th partial sum
s
n
.
Show that

s

s
n

6
a
n
+1
.
3.
Consider the series
1

1
2

1
3
+
1
4
+
1
5

1
6

1
7
+
···
where the signs come in pairs. Does it converge?
4.
Investigate the convergence of the following series.
(a)
∞
X
n
=1
(

1)
n
n
p
, where
p
is some real number.
(b)
∞
X
n
=1
(

1)
n
√
n
n
+ 10
.
(c)
∞
X
n
=1
x
n
√
n
, where
x
is some real number.
5.
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 Winter '10
 Spyros

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