Confusion around the way infinite series are defined, and how they are used

Question

According to Wikipedia (https://en.wikipedia.org/wiki/Series_(mathematics)), what we call the sum of an infinite series is really just a shorthand for its limit. E.g. the sum of $1/2+1/4+1/8+ ...$ is 1. This is the number that the series tends to, the smallest number that the partial sums will never reach, no matter how many terms you add together. Conceptually, this seems much easier to grasp than actually "adding all the terms up". However, this seems to be in opposition with other ideas in maths. For example, 0.999... is said to be a complete number (one of the main misconceptions about it is that it is growing or expanding). But doesn't this imply that you can "add all the terms up", contrary to the careful way the sum of an infinite series is defined? By "terms" I mean this (I have often seen this definition of 0.999.. in the proofs that it is equal to 1): $$\frac {9}{10}+\frac {9}{100}+\frac {9}{1000}+\frac {9}{1000}+...$$However, I am wary of defining it in this way because it seems like there is no way to truly assigning a value to this series — the best you can do is say that it's limit is 0.999..., which seems different to saying that it is 0.999... This is similar to how I don't quite follow the leap made in the resolution of Achilles and the Tortoise, where the limit of $1/2+1/4+1/8+1/16+...$ is 1, so consequently the tortoise can be caught up. I feel as if the concept of making "infinite steps" is strange, especially when it seems like the definition of an infinite series has circumvented this problem with the idea of limits. I'm sorry if this question sounds like it has turned into many; I'm just genuinely curious about infinity and the role it plays in mathematics. Thanks

Answer 1

We define $0.999\dots$ as the limit of this geometric series, which of course is $1$. This trips up a lot of people, but it shouldn't. It makes just as much sense to write$$0.999\dots=\sum_{k\ge1}\frac{9}{10^k}=\lim_{n\to\infty}\sum_{k=1}^n\frac{9}{10^k}=1$$in base 10 as to write$$0.111\dots=\sum_{k\ge1}\frac{1}{2^k}=\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{2^k}=1$$in binary.