We seek closed form for the power series , where , and . First we note that the ratio test yields
, where is the golden ratio. Thus the series converges on the region of the complex plane where . Now let the power series be denoted by , and note that
and so . It seems so unlikely, before you know better at least, that things like that can be done. Why should it be that the nth Fibonacci number is given by , the integral along some closed contour in the complex plane of a function that otherwise seems unrelated. That’s maths!