My PhD advisor wrote a paper titled "when is 0.999... equal to 1": https://www.math.upenn.edu/~pemantle/papers/nearly.pdf https://www.math.upenn.edu/~pemantle/p

My PhD advisor wrote a paper titled "when is 0.999... equal to 1": https://www.math.upenn.edu/~pemantle/papers/nearly.pdf https://www.math.upenn.edu/~pemantle/papers/nearly.pdf (It wasn't, in the case he was interested in - he had a sum that he suspected might be 1, from numerical evidence, but that actually turned out to be a combination of zeta values equal to 0.999222...)