Writing this concatenating decimal as a series/fraction

Question

I am trying to work out a way to write the number $0.123456789101112...$ as a series, I thought at first doing something like: $$\sum_{n=1}^N\frac{n}{10^n}$$ but when we get into the double digits this doesn't work, so how should I do this? Also is there a way to evaluate sums of the form above? Thanks

Answer 1

This number is known as the Champernowne constant, and is known to have a few summation identities, neither of which are pleasant:

$$\sum_{n=1}^\infty\frac n{10^{~\left(\sum\limits_{k=1}^n\left\lceil\log_{10}(k+1)\right\rceil\right)}} \qquad \sum_{n=1}^\infty\frac n{10^{\left(n+\sum\limits_{k=1}^{n-1}\left\lfloor\log_{10}(k+1)\right\rfloor\right)}}$$

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Also is there a way to evaluate sums of the form above?

An infinite sum of the form you describe ($\sum n/10^n$) can be evaluated by taking derivatives of the geometric series. Recall that

$$\sum_{k=0}^\infty x^k = \frac{1}{1-x}$$

whenever $|x|<1$. Take the derivative with respect to $r$ of both sides -- you can do it on the left side term by term due to absolute convergence -- and then multiply by $x$. You get

$$\sum_{k=1}^\infty kx^k = \frac{x}{(1-x)^2}$$

In your case, $x=1/10$.