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In-Depth Information
u
v
both
and
are divergent, and the partial sums of both series
tend to infinity.
Rearrangements
In qn 62 weproved that 1
. . . was
convergent.
If terms of this series are rearranged so as to have two positive terms
preceding each negative term, we get the following result.
1.3
1.2
1.1
1
0.9
n
5
10
15
20
(1
)
(
)
(
)
...
(1
)
(
)
(
)
...
(1
)
(
)
(
)
(
)
(
)
(
)
...
(1
)
(
)
(
)
...
(
)
(
)
(
)
...
(1
)
(
)
(
)
...
(1
)
(
)
(
)
...
(1
...)
(1
. . .).
This is apparently
times the original series, before rearrangement!
72 For the series
a
1
1
. . . find the
partial sum
s
of
thefirst 2
n
1 terms. Deduce that the series is convergent and find
its sum. Find an expression for
a
of thefirst 2
n
terms, and the partial sum
s
and for
a
.
