Can Infinitely Reordered Products Converge to Different Values?

[mmzaj]

does the following infinite product converge ? and what to ?

$$\prod^{\infty}_{n=1}(1-\frac{x^n}{n})$$

i know it has something to do with elliptic functions ...

 

[jambaugh]

I'm not sure except I believe so for |x|<1.
You can apply series test to the infinite sum you get by taking the logarithm of the infinite product.

Consider also the power series expansion:
$$\ln \left(1+z\right) = \sum_{k=1}^{\infty} (-)^{k+1}z^k$$
for $|z|<1$.
So you can express the logarithm of your product as an infinite sum of power series.
See if reordering gives you an answer you can use.

 

[adriank]

Be careful with reordering! If a series like that only conditionally converges, then by reordering, the series can converge to a different value (or even diverge).

If it helps, Mathematica is unable to simplify that product, except for trivial values of x.