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Let $f_n(x)$ be defined as the $n$th digit of the number $x$.

The result of $f_n(x)$ can be only ${0,1,2,3,4,5,6,7,8,9}$ for base 10.

For example, if $x=12.46$, then

$f_2(x)=0$;$f_1(x)=1$;$f_0(x)=2$;$f_{-1}(x)=4$; $f_{-2}(x)=6$ ; $f_{-3}(x)=0$.

If we have such function , we can write any real number easily as shown below:

$x=\sum \limits_{n=-\infty}^\infty f_n(x) 10^n$

I tried to find power series expression of the function. $f_n(x)=a_0(n)+a_1(n)x+a_2(n)x^2+\cdots$

$$\begin{align*} x&=\sum \limits_{n=-\infty}^\infty f_n(x) 10^n\\ &=\sum \limits_{n=-\infty}^\infty (a_0(n)+a_1(n)x+a_2(n)x^2+\cdots ) 10^n\\ \sum \limits_{n=-\infty}^\infty a_0(n) 10^n&=0\\ \sum \limits_{n=-\infty}^\infty a_1(n) 10^n&=1\\ \sum \limits_{n=-\infty}^\infty a_2(n) 10^n&=0 \end{align*}$$

But this do not give me so many thing to define $a_k(n)$

Is it possible to find $a_k(n)$ with some method that known?

I also wonder what the function properties of $f_n(x)$ are? (such as $f_n(x+y)$, $f_n(x.y)$ etc.) I wonder the literature about the function.

Could you please share your knowledge about the function? Sorry for your time if It was asked before or very basic for number theory.

Thanks a lot for advices and answers

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Maclaurin series expansion (which is what you are doing) approximates the behavior of the function in a small interval around 0. But $f_n(x)$ is identically 0 in the interval $[0, 10^{-n}]$, so your Maclaurin expansion will have $a_i = 0$ for all $i$. Power series are not going to work here. –  MJD May 6 '12 at 20:06
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1 Answer

up vote 2 down vote accepted

As @Mark Dominus said you won't be able to solve for your $a_k(n)$, but you can find a Fourier series for $f_n$.

First $f_n(x) = f_0(10^{-n}x)$ and $x = \sum 10^n f_0(10^{-n}x)$ so I will only deal with $f_0$.

$f_0(x+10) = f_0(x)$, so let us extend $f_0$ to the negative numbers by $f_0(x-10)=f_0(x)$ so $f_0$ is periodic over all $\mathbb{R}$. For cleanliness let's also define $$f_0(k)=\lim_{\epsilon\to 0}\frac{f_0(k-\epsilon)+f_0(k+\epsilon)}{2}$$ at the round integers $k$, so e.g. $f_0(2.9999\ldots) = f_0(3.0) = 2.5$ and $f_0(29.9999\ldots)=f_0(30.0)=4.5$. Also let $g_0(x) = f_0(x)-4.5$, then $g_0$ is an odd periodic function and has a Fourier sine series. In fact $g_0$ is the difference of two sawtooth waves. It's fairly straightforward to find $$ g_0(x) = -\frac{10}{\pi}\sum_{k=1}^\infty b_k \sin\left(\frac{k\pi x}{5}\right) $$ where $$ b_k = \begin{cases}0 & \mathrm{if}~10\mid k\\ 1/k & \mathrm{otherwise}\end{cases} $$

I don't know of any literature about this function. I realize this hasn't answered your specific questions, but I hope it's of some interest.

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Really you shew very interesting results about the function.Thanks a lot. Can I find correctly nth digit of $\pi$ ,$e$ with your formula without calculating other digits. Can It be used for finding record digits of any irrational number? Thanks for advice. –  Mathlover May 18 '12 at 10:31
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