I previously saw that peculiarly $\frac{987654321}{123456789}\approx 8$. I was wondering if there was any significance to it i.e. if there is any way to derive this approximation (aside from long division).
I already shared my solution. I would love to see other interesting methods (if any).
On
We can express $987654321$ as approximately $\sum_{k=-9}^\infty -k\cdot 10^{-k-1}$ and $123456789$ as approximately $\sum_{k=1}^\infty k\cdot 10^{9-k}$
If we take $$f(x)=\sum_{k=-9}^\infty x^k=\frac{x^{-9}}{1-x}$$ then $$f'(x)=\sum_{k=-9}^\infty kx^{k-1}=\frac{-9x^{-10}+10x^{-9}}{(1-x)^2}$$ $$\implies f'(x)=\sum_{k=-9}^\infty kx^{k-1}=\frac{-9x^{-10}+10x^{-9}}{(1-x)^2}$$ $$\implies x^2f'(x)=\sum_{k=-9}^\infty kx^{k+1}=\frac{-9x^{-8}+10x^{-7}}{(1-x)^2}$$ $$\implies -10^2\cdot f'\left(\frac{1}{10}\right)=\sum_{k=-9}^\infty -k\cdot 10^{-k-1}=\frac{8\cdot 10^{10}}{81}$$ Similarly, if we take $$g(x)=\sum_{k=0}^\infty x^k=\frac{1}{1-x}$$ then $$g'(x)=\sum_{k=1}^\infty k\cdot x^{k-1}=\frac{1}{(1-x)^2}$$ $$\implies x^{-8}g'(x)=\sum_{k=1}^\infty k\cdot x^{k-9}=\frac{x^{-9}}{(1-x)^2}$$ $$\implies 10^8\cdot g'\left(\frac{1}{10}\right)=\sum_{k=1}^\infty k\cdot 10^{9-k}=\frac{10^{10}}{81}$$
Hence, from our approximations for $987654321$ and $123456789$, it follows that $\frac{987654321}{123456789}\approx 8$
More generally, in base $b$, the number with digits decreasing from $b-1$ to $1$ is $$ N = \sum_{i=0}^{b-2} (i+1) b^i = \frac{b-2}{(b-1)^2} b^{b} + \frac{1}{(b-1)^2}$$ while the number with digits increasing from $1$ to $b-1$ is $$D = \sum_{i=0}^{b-2} (b-1-i) b^i = \frac{b^b}{(b-1)^2} - \frac{b^2-b+1}{(b-1)^2}$$ For large $b$, the dominant terms are those with $b^b$, so $$ \frac{N}{D} \sim b-2 $$