Question: We know it converges for any prime $p$. I just want to know how to compute its exact value:
$$\prod\_{n=1}^{\infty} (1-p^{-n})$$

Answer: 
Let's fix the issue of giving bounds on the infinite product. The [pentagonal number theorem](https://en.wikipedia.org/wiki/Pentagonal_number_theorem) gives, after grouping pairs of consecutive terms with the same sign, an [alternating series](https://en.wikipedia.org/wiki/Alternating_series_test) with terms that are decreasing in modulus. So for instance for $c:=\prod \_ {n\ge1} (1-2^{-n})$ 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 $1/4$.
Wolfram MathWorld gives the expression of this product in terms of the q-Pochhammer symbol and the Jacobi theta function. See formulas (46) and (47) in

<https://mathworld.wolfram.com/InfiniteProduct.html>