 # Three interesting infinite series

Is quite easy: let S be the total sum of the series. If you divide by 2 every term you get 1/2+1/4+1/8 etc, i.e. S - 1. So
S/2 = S - 1 - > S/2 = 1 - > S = 2
S/2 = 1

This can be generalized for every series of 1/n^m. It can be demonstrated that the sum converge if and only if m is > 1. The first series ha m = 1 and does not converge.

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I watched it, was just making a joke 1 Like

I get the first one now. By definition every nth term will be halfway between the last point and 2, so the series will never exceed 2. With infinite terms it becomes 2.

Math is hard for me.

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Surely, you mean that he quantum tunneled the last plank length.

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I used to have a T-shirt with that cartoon on the front 2 Likes

Interesting.

Did you know that the infinite sum of all positive integers 1+2+3+4+… is equal to -1/12?

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We’ve been over that before. Looking at it as a Dirichlet series, and then extending the zeta function by its analytic continuation, it “equals” that, yeah, sorta. This explains the whole story: https://plus.maths.org/content/infinity-or-just-112

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Yeah, soon after I posted that I found a video explaining how the usual demonstration had been debunked. It concluded that it had still some validity but using “equals” would be an exaggeration.

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