I'm trying to find a pretty closed form for:

$$0.9 \cdot 0.99\cdot 0.999 \cdots=\prod_{n=1}^\infty \frac{10^n-1}{10^n}$$

The Nth partial product can be expressed as:


But it seems impossible to generalize the expansion of the numerator polynomial in 10.

Wolfram Alpha computes the following approximation:


Hence I strongly believe that a closed form exists.


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