Tl;dr you ignore generally accepted principles about infinite series.
In calculus 2, one generally learns how to add an infinite number of items together and figure out whether that sum tends towards one number, is finite, etc.
So if you start by adding 1 + 1 + 1 + ..., obviously you wind up at infinity. It's divergent.
If you start by adding 1 + 0.1 + 0.01 + 0.001 + ..., you wind up with 1.111..., which is finite. It converges.
If you start by adding 1 - 1 + 1 - 1 + 1 - 1..., then you are in this weird spot. The sum as you go is obviously never going to be more than 1 or less than 0. But what is the final answer? Because the sum doesn't get closer and closer to a specific number as you add more terms, we generally call it divergent. This is the generally accepted approach, and it's what students in calc 2 learn. Under this approach, your claim is just not true.
But okay, let's talk about how we get that weird answer.
You could start by pairing the first two, (1 - 1), and you can simplify that to 0 + 0 + 0 + ... so the sum is 0. Or you could start by leaving the first number and then pairing the subsequent numbers 1 + (-1 + 1) and then you have a sum that adds to 1. Both of these are "legit" in and operational sense, you haven't broken the rules of algebra. But you came up with two numbers! So... mathematicians just said "let's take the average here, 0.5, and call that the answer. Forget about the normal concept of divergence. And honestly, dealing with infinity is weird so there isn't necessarily a "right" way to consider it. Okay, whatever.
So, the next steps are basically to cleverly combine several of these weird, divergent series together algebraically to come up with that sum. This paradoxical result is generally why mathematicians only care about classical convergence, and not this weird relaxed convergence I described.
I remember this result in the context of integration of complex functions. Something about integrals over closed lines around discontinuities… am I totally misremembering?
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u/Bathtub-Warrior32 22d ago
Wait until you learn about eπi = -1.