Question

Geometric series using complex numbers with a length of 1.

Answer 1

So when coming across the sum: \[S=1-1+1-1+1-1+...\] I noticed we could consider it kind of like a geometric series, so we'd have \[S= \sum (-1)^n\] and the formula for a geometric series looks like this: \[\frac{1}{1-x}\] So if you plug in -1 here for ours we get: \[\frac{1}{1-(-1)}=\frac{1}{2}\] which is also kind of like the average point we are going between here from 0 and 1.

So what if we did that with another number, like a complex number, say, i? That'll make a box like this every time we add it up so \[S=i^0+i^1+i^2+i^3+i^4+...\] \[S=\frac{1}{1-i}=\frac{1}{2}+\frac{i}{2}\] which also in the middle!

So is this a general thing or what? How would I go about showing this?

Answer 2

On a somewhat related note, is this also the same thing as when a piecewise continuous function converges to the average of the two points in a Fourier Series?