Graphics Reference
In-Depth Information
5
Series
Infinitesums
Preliminary reading:
Cohen, D., Northrop ch. 7, Nelsen, R. B.
Concurrent reading:
Ferrar.
Further reading:
Rudin, ch. 3, Knopp.
Sequences of partial sums
1 Criticisethefollowing argument:
if
S
1
x
x
x
...,
then
xS
x
x
x
x
...,
so
S
xS
1,
1
1
x
.
and therefore
S
Try putting
x
2!
If the argument in qn 1 were sound we could put
x
1 and obtain
the sum of the series
1
1
1
1
1
...
to be
. But the same series could be plausibly thought to have a sum
of 0
—
(1
1)
(1
1)
(1
1)
...
—
or a sum of 1:
1
(
1
1)
(
1
1)
(
1
1)
...
These paradoxical conclusions show that great care must be used in
arguing with infinitesums. Theordinary associativelaw for finitesums,
a
(
b
c
)
(
a
b
)
c
, does not lead to a clear answer.
