In class today I was looking at the sum $1 +\frac{1}{2}+\frac{1}{3}+\frac{1}{4}...$ and with a bit of fiddling, managed to come up with the following: $$\sum_{n=2}^\infty \left(\sum_{m=2}^\infty \frac{1}{n^m} \right) = 1$$ I managed to show this in 2 different ways: Method 1 and Method 2, mostly the same method, yes.
But I was wondering if this is at all meaningful, has it been used in anything at all? Does it mean anything apart from it's something that's pretty cool?
It is one of the several identities involving the Riemann Zeta Function $$\sum_{n=2}^{\infty}(\zeta(n)-1)=1.$$ Here there is another one $$\sum_{n=2}^{\infty}(-1)^n\cdot (\zeta(n)-1)=\frac{1}{2}.$$ You can try to prove it by considering $$\sum_{n=2,4,6,\dots}(\zeta(n)-1)=\frac{3}{4}$$ and $$\sum_{n=3,5,7\dots}(\zeta(n)-1)=\frac{1}{4}.$$