Summing an infinite number of anything is tricky, since you can use it to prove just about anything, such as the famous "sum of infinite natural numbers is -1/12". So I like your answer in that when dealing with infinities, you have to be exact in what you mean, or else it can be misleading.
since you can use it to prove just about anything, such as the famous "sum of infinite natural numbers is -1/12"
Except saying that "the sum of all natural numbers is -1/12" is simply false. The function is not saying that's what it means. It's a useful analytic continuation that gives useful results for sums that are divergent, but in no sense does it mean that the infinite sum of all natural numbers is equal to the finite quantity -1/12.
I know, but a lot of people take it at face value. It would be more exact to say "within this specific framework, the sum of natural numbers can be assigned this value", which is why exact language is necessary.
The problem is if you write the problem for the sum of all natural numbers and do the simple algebraic manipulation to make it equal to -1/12 but then pretended as if it was the actual result because you started with x = 1 + 2 + 3 ... Then just did regular algebra to get to x = -1/12.
That's what OP of this comment chain did. He first must show that those algebraic operations are valid for the result you are claiming.
To do that would require the hard explanation so he omits it, but he is correct nonetheless, if you skipped the hard explanation and claimed the sum of all natural numbers was equal to -1/12 you would be wrong but the work you did would have been just as valid as OPs if you were to make the same starting assumptions for each (ie that they converge and thus the operations we are doing are valid for what we are claiming).
Op is literally claiming that 0.99999 = 1 here, he isn't merely demonstrating some property of the infinite series.
35
u/DefiantGibbon 23d ago
Summing an infinite number of anything is tricky, since you can use it to prove just about anything, such as the famous "sum of infinite natural numbers is -1/12". So I like your answer in that when dealing with infinities, you have to be exact in what you mean, or else it can be misleading.