When would it be useful to assume a divergent series is equal to something?

So I was revisiting this Numberphile video:


because I feel like people gave Numberphile a lot of shit for it and it’s corollary video about the sum of the natural numbers equaling -1/12.

So he begins the video by saying what does 1-1+1-1+… equal too, which subtlety implies that it has to equal something. I thought that was a weird assumption considering the fact that not everything is equal to something, like 5÷0 isn’t equal to something.

I think this is where people arrive at and will say divergent series don’t equal anything. But one of the fantastic things about math is you may choose your assumptions to be whatever you want (whether people agree or not), what’s more important is if it is helpful to assume those assumptions.

So, I was wondering if anyone could think of a senario where giving a value to a divergent series would be helpful.

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Nevin Manimala

Nevin Manimala is interested in blogging and finding new blogs https://nevinmanimala.com

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