structural:reformatting, structural:summarization, information:classification, linguistic:translation
A more complicated example

Consider this power series:

where the rational numbers B are the Bernoulli numbers. It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series. But the theorem of complex analysis stated above quickly solves the problem. At z = 0, there is in effect no singularity since the singularity is removable. The only non-removable singularities are therefore located at the other points where the denominator is zero. We solve

by recalling that if z = x + iy and e = cos(y) + i sin(y) then

and then take x and y to be real. Since y is real, the absolute value of cos(y) + i sin(y) is necessarily 1. Therefore, the absolute value of e can be 1 only if e is 1; since x is real, that happens only if x = 0. Therefore z is purely imaginary and cos(y) + i sin(y) = 1. Since y is real, that happens only if cos(y) = 1 and sin(y) = 0, so that y is an integer multiple of 2 π. Consequently the singular points of this function occur at

The singularities nearest 0, which is the center of the power series expansion, are at ±2 π i. The distance from the center to either of those points is 2 π, so the radius of convergence is 2 π.