4
$\begingroup$

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:

$$\frac{(10-1)(10^2-1)\cdots(10^N-1)}{10^{\frac{n(n+1)}{2}}}$$

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

Wolfram Alpha computes the following approximation:

$≈0.89001009999899900000010000999999998999990000000000100000099999999999989999999000000000000001000000009999999999999999899999999900000000$

Hence I strongly believe that a closed form exists.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.