Sum of powers of i

James Tanton asks:

What is ?

Of course it’s , right, by the usual formula for summing a geometric series? But this says that

when .  And , so it doesn’t work here. But who cares? Start taking partial sums. The sum is (after simplifying using ):

and we can write down partial sums: — and the average of this series is $(1+i)/2$, which is $1/(1-i)$. It’s a complex version of Grandi’s series (, and indeed the argument I’ve outlined here is Cesaro summation.)