Consider the following expression: $$\prod_{k=1}^{\infty} \sum_{\ell =1}^{k} \frac{9}{10^\ell} = 0.9\times0.99\times0.999\times0.9999\times...$$
I have a curiosity about the decimal expansion of this number. WolframAlpha describes this as $\left(\frac{1}{10};\frac{1}{10}\right)_{\infty}$, where the subscript $\infty$ denotes the Pochhammer symbol. Maybe I'm too tired to figure this out, or something, but I cannot exactly see why the decimal expansion appears as it is. Here's a bunch of the digits:
$$0.890010099998999000000100009999999989999900000000001000000999999999999899999990000000000000010000...$$
The pattern here is, for the $8's$ and $9's$, is to surround an $8$ by four nines on the left and three nines on the right. Then eight nines on the left and five on the right. Then twelve of them and seven of them, etc.
For the zeros and ones, surround a $1$ with two zeros on each side, then again with six zeros on the left and four on the right. Then ten zeros on the left and six on the right. Same pattern, but different starting point.
It alternates the $8,9$ pattern with the $0,1$ pattern. I described the pattern in a really long way but I'm not sure how else to explain it without being ambiguous.
Why does the decimal expansion have this form...?
In general, $\,(q;q)_\infty = 1 - q - q^2 + q^5 + q^7 - \dots\,$ which is explained in the Wikipedia articles Euler function, $q$-Pochhammer symbol, and Pentagonal number theorem. In your particular case, $\,q=.1\,$ which explains the decimal expansion you are asking about since some powers of $\,q\,$ are added and some subtracted.