Recently we talked about how Euler managed, through some magic tricks, to find that the sum of the inverse of the square numbers is … That was a piece of *virtuosismo*, but even geniuses sometimes slip up. This one is funny. You may know the sum of a geometrical series:

(There are many ways to understand that formula, I’ll give you a nice one soon). Now, using the same formula, sum the *inverse powers*:

So far, so good. Now, imagine that we want the sum of *all* powers, positive and negative:

Amazing!!! The sum of all powers is zero!! Nice, ein? Euler concluded that this proved the possibility of the creation of the world from nothing, *ex nihilo*. So, this is easy… what is the problem with this proof?

BTW, I read this story and many others in William Dunham’s book The master of us all…

I'm better than you all even when I'm wrong!

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What is that power that results in -1?

None. But the 1, which is x

^{0}is countedtwice, both in positive and negative powers, so you have to substract it :)The summation you used is only valid for -1<x<1. You cannot simultaneously have -1<x<1 and -1<1/x<1 fit this criteria so the summation is not valid.

Good, Hologram0110!! :) So, the summation is *never* valid but… it’s intriguing nonetheless, isn’t it? ;)