| It is unclear what one wants by the 'value' of this product...
In terms of a decimal expansion, I claim the value should look like this:
0 0.89
1 00 1 00 9999 8 999
2 000000 1 0000 99999999 8 99999
3 0000000000 1 000000 999999999999 8 9999999
...
k <4k-2 0's> 1 <2k 0's> <4k 9's> 8 <2k+1 9's>
It's almost surely transcendental.
As a function of a complex variable,
f(z) = PROD(k>0) (1-1/z^k) = PROD(k>0) (z^k-1)/z^k
is interesting since it has an essential singualarity at zero, and
zeroes on the unit circle. This may give a closed form expression in
terms of some known function.
Note that you can rearrange the product into the form
log(f(z)) = - SUM(k>0) 1/(k*(z^k-1))
at least where this converges. It doesn't seem to telescope though.
- Jim
|
| I should explain how to obtain the decimal expansion in .1 - it comes
from plugging into a well known identity involving the generating
function for the number of partitions of an integer, due to Euler.
This is in books on number theory, such as Hardy and Wright. See also
Knuth, "The Art of Computer Programming".
The identity is
f(z) = PROD(k>0) (1-z^k)
= 1 + SUM(k>0) (-1)^k * [z^(k*(3*k-1)/2)+z^(k*(3*k+1)/2)]
With z = 0.1 you can easily fill in a decimal expansion.
The generating function is actually the reciprocal of f(z),
F(z) = SUM(k>=0) p(k)*z^k
= 1 / f(z) = 1 / PROD(k>0) (1-z^k)
where p(k) = the number of ways of partitioning an integer.
Eg. p(5) = 7 since 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1,
7 ways. There are tables of partions and some other combinatorial
stuff in AMS55, along with some (hairy) closed form expressions for
p(n).
My comment about the singularity of the f(z) in .1 is probably meaningless
since it doesn't converge when z < 1.
It's doubtful if there is a simpler way of expressing the sum than this.
However, products like f(z) are known as theta functions and are actually
useful since they converge quickly and by taking ratios of them one
can get the values of elliptic functions. The latter are nice for
practical EE work like conformal mapping, synthesizing filters and
phase splitters, and of course making pretty pictures of minimal
surfaces.
- Jim
|