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between 1 and 2. Let 1
2. Let
s
denote the sum of the first
n
terms of the series
1
2
1
3
1
4
...
1
n
....
1
n
n
Verify that
s
s
.
Write down these inequalities for
n
2, 4, 8, . . ., 2
and add them to
show that
1
r
1
2
2
4
4
2
2
1
(1/2
)
1
...
,
(1/2
)
using qn 2 for theequality.
Show that
1 implies 1/2
1.
Deduce that the partial sums of
arebounded and so the
series is convergent. This proof holds for all real
1.
1/
r
Cauchy's nth root test
33 Usethefact that
n
/(2
n
1)
and thefirst comparison tst (qn
(
n
/(2
n
26) to prove that the series
1))
is convergent.
34 If
n
3, provethat
4
n
5
n
36
43
,
2
and using thestart rule(qn 16) and thefirst comparison tst (qn 26)
deduce that the series
(4
n
/(5
n
2))
is convergent.
0
k
ε
kk +
ε
1
35
Cauchy
'
s nth root test
To generalise the results of qns 33 and 34, we suppose that we have
a series of positive terms,
, for which the sequence (
a
a
k
as
)
n
, and that 0
k
1. By choosing
(1
k
), show that for
suMciently large
n
,
a
1. Deduce
from thestart rule(qn 16) and thefirst comparison tst (qn 26) that
where
k
(1
k
)
a
is convergent.
36 Apply Cauchy's
n
th root test to the series
n
/2
(comparewith qn
10), and to the series
n
(0.8)
.