Pi approximation

Question

If $d(a,b)=$ largest $n$ such that $a$ and $b$ agree on all digits upto $n$. Eg. $d(\pi,3.14)=3$, $d(0.1234667,0.1234669)=7$. What is the asymptotics of $d(\pi/4,1-1/3+1/5-1/7+\cdots(\pm)1/m)$ as $m\rightarrow\infty$?

Answer 1

Using the standard estimates for an alternating series of decreasing terms [1], we know the error $$ \frac{\pi}{4} - \sum_{k=1}^m \frac{(-1)^{k+1} }{2k-1} = \sum_{k=m+1}^{\infty} \frac{(-1)^{k+1} }{2k-1}$$

has magnitude $\sim \displaystyle \frac{1}{2m}.$ The number of agreed digits is thus asymptotic to the number of leading decimal zeros in the decimal expansion of $ \displaystyle \frac{1}{2m},$ which is $ \sim \log_{10} 2m .$