Imagine that'd I'd like to investigate the digits of $\sqrt{2}$, or of any real number. If I want a formula for the nth digit of a real number $x$, we have,
$$(1) \quad \operatorname{d_n}(x)=\lfloor 10^{n-1} x \rfloor -10 \cdot \lfloor 10^{n-2} x \rfloor$$
Where $\lfloor y \rfloor$ denotes the floor function. However, I wish to be able to do more than just have a "formula". I wish for it to in some way, illuminate/pick-out the aspects that I want and destroy the information I don't want.
If I try expanding $(1)$ using a sine series, I get,
$$(2) \quad \operatorname{d_n}(x)={9 \over 2}+{1 \over {\pi}} \cdot \sum_{k=1}^{\infty} \left[ {1 \over k} \cdot \left( {\sin(2 \cdot 10^{n-1} \cdot \pi \cdot x \cdot k)-10 \cdot \sin(2 \cdot 10^{n-2} \cdot \pi \cdot x \cdot k)} \right) \right]$$
Is there anything simpler? Specifically, are there formulas that don't assume knowledge of the digit expansion of $x$?
I know that if a number can be written as,
$$(3) \quad x=\sum_{n=1}^{\infty} b^{-n} \cdot R(n)$$
And $R(n)$ goes to $0$ in the limit, then we can expand using $\{b^n \cdot x \}$ to pick out digits, but this is only useful if we have $b=10$. Plus it assumes that we neglect terms after $R(n)$ becomes "small" enough.