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73 For the series
b
1
1
...
1/(2
n
1)
1/2
n
1/
n
...
comparethepartial sum of thefirst 3
n
terms with the partial sum
of thefirst 2
n
terms of the series in qn 62. Deduce with the help of
qn 65 that this series has the sum log 2. Find an expression for
b
,
b
, and for
b
.
74 Provethat thest
a
n
N
of qn 72 is identical to the set
b
n
N
of qn 73.
Deduce that a rearrangement of the terms of a conditionally
convergent series may alter its sum.
75 (
Dirichlet
, 1837)
(i) Write down the first nine terms of the series whose (3
n
2)th
term is 1/
(4
n
3), (3
n
1)th term is 1/
(4
n
1) and 3
n
th
term is
1/
(2
n
).
(ii) Check that this series is a rearrangement of
(
1)
/
n
.
(iii) Is
(
1)
/
n
convergent?
(iv) Provethat, if
n
12, then
1
n
1
1
1
3
1
2
n
.
4
n
4
n
(
Hint
. Multiply through by 2
n
.)
(v) Prove that the series in (i) is divergent.
So a conditionally convergent series may be rearranged to diverge!
With the result of qn 71 it is possible to see how to rearrange the terms
of any conditionally convergent series, to produce another series which
converges to any limit we please, say,
l
. First take
positive
terms until
l
is just exceeded, then take
negative
terms until the sum is just less than
l
, and so on. The divergence of
u
and
v
means this is always
possible, and the fact that both (
u
) are null sequences
guarantees the convergence. (
Riemann
, 1854.)
) and (
v
76 Let
a
be a sequence of
positive
terms with sum
A
and let
b
be
a rearrangement of the same series. Let
s
a
and let
t
.
Supposethat thefirst
N
terms of the sequence of
b
s areincluded
within thefirst
M
terms of the sequence of
a
s. Provethat
t
b
s
A
. Deduce that
b
is convergent to a sum
B
(say) and
that
B
A
.
