# Summing an infinite series

I dare say what I am about to write is no great revelation and indeed I was probably taught this for ‘A’ level maths, but forgot it…

Anyway, I was reading about the Fast Fourier Transform (FFT), which, discovered in the 1960s, radically improved the speed of signal analysis. Older algorithms had been able to analyse $N$ signal samples in $O(N^2)$ time, while the FFT manages it in $O(Nlog_2N)$ time.

It does this by relying on the fact that one can combine two $N$ samples to get a $2N$ sample in $N$ operations, ie., $N \rightarrow 2N$ in $N$ operations $2N \rightarrow 4N$ in $2N$ operations $4N \rightarrow 8N$ in $4N$ operations

Hence we can get $N$ samples in $log_2N$ doublings. But how many operations does that take?

For the above example ( $N = 8$) we can see it is $\frac{N}{2} + \frac{N}{4} + \frac{N}{8}$ and for the more general case we can see this is $N(2^{-1} + 2^{-2} + ... )$ $N$ times an infinite series.

For the overall complexity of the algorithm to be $O(Nlog_2N)$ this series should add to 1. Does it?

Yes, it does, and here’s a proof…

We can rewrite our series as… $X = r + r^2 + r^3 + r^4 + ...$

Then $Y = \frac{X}{r} = 1 + r + r^2 + r^3 + ...$ $= 1 + X$

Hence $\frac{X}{r} = 1 + X$

And $\frac{X}{r} - X = 1$.

As we know $r = \frac{1}{2}$ we get $2X - X = 1$ so $X = 1$