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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)
.
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