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n
.
What is the partial sum of the first hundred terms of the series
a
(
n
n
, and let
b
28 Let
a
1)
1/
?
(i) Useqn 25 and qn 3.20 to provethat
a
is divergent.
1/(
(
n
1)
n
).
(iii) Provethat 0
a
(ii) Provethat
a
.
(iv) Use the first comparison test, qn 26, and the scalar rule, qn
19, to provethat
b
b
is divergent.
So, although (
b
is divergent. Rather
disconcertingly, this establishes that the converse of qn 11 (the null
sequence test) is false. So do not fall into the trap of thinking that
b
) is a null sequence,
b
must be convergent when (
b
) is null.
29 Usethefirst comparison tst (qn 26), and qns 27 and 28 to show
that
1/
n
is convergent when
2 and divergent when 0
.
The harmonic series
30 For the series
1
2
1
3
1
4
...
1
n
...,
1
let
s
denote the sum of the first
n
terms. For each
n
, show that
n
2
n
1
2
.
s
s
Write down these inequalities for
n
1, 2, 4, 8, . . ., 2
, and add
them to prove that
1
r
1
n
.
Is
1/
n
convergent or divergent?
1/
n
31 For what values of the positive real number
The convergence of
can you besurethat
the series
1/
n
is divergent? Just use the first comparison test (qn
26) and qn 30 at this stage.
32 In qn 29 we proved that the series
1/
n
was convergent when
2, and in qn 31 we saw that this series is divergent when
1.
We have not yet examined the convergence of
1/
n
for values of
