We know it converges for any prime p. I just want to know how to compute its exact value: $$\prod_{n=1}^{\infty} (1p^{n})$$
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Wolfram MathWorld gives the expression of this product in terms of the qPochhammer symbol and the Jacobi theta function. See formulas (46) and (47) in 


Let's fix the issue of giving bounds on the infinite product. The pentagonal number theorem gives, after grouping pairs of consecutive terms with the same sign, an alternating series with terms that are decreasing in modulus. So for instance for $c:=\prod _ {n\ge1} (12^{n})$ one has $$1\frac{1}{2}\frac{1}{4}+\frac{1}{32}+\frac{1}{128} \frac{1}{4096}\frac{1}{ 32768 } < c < 1\frac{1}{2}\frac{1}{4}+\frac{1}{32}+\frac{1}{128}$$ that is $ 0.288787842 < c < 0.2890625, $ in any case larger than $1/4$. 

