Minimum and maximum of the following summation series

Question

So, the thing reduces to $$0.53+\frac{a_3}{10^3}+\frac{a_4}{10^4}+\frac{a_5}{10^5}+...$$ If $a_3,a_4,...=0$ then it gives $0.53$.

What about the upper bound? Does it happen when $a_3,a_4,...=9$?

Answer 1

An observation: the summation coincides with the number

$$0.53a_3a_4a_5\cdots$$

Since $a_k$ can be any digit from $0$ to $9$ from all $k$, the conclusion is obvious: we maximize the number if $a_k = 9$ for all $k$, and minimize it if $a_k = 0$ for all $k$.

A caveat: all rational numbers that can be written in the form $a/2^b5^c$ for integers $a,b,c$ have two decimal expansions in base $10$, one ending in a tail of $0$'s and another in one of $9$'s. This ties into the whole "is $0.999 \cdots = 1$?" dilemma you might find discussed elsewhere on the site, if you're not familiar with it. We have to account for this, technically.

Thus in turn, the maximum is $0.539999\cdots = 0.54$ and the minimum is $0.530000\cdots = 0.53$.

Answer 2

Indeed. It's easy to convince yourself that $0.009 + 0.0009 + 0.00009 + \ldots = 0.01$ so the upper bound is $0.54$.