Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 6 / ii / Solution

The coefficients in the Taylor series areFor either series, the ratio test gives absolute term ratio $|x|/(n+1)$ or $2|x|/(n+1)$, tending to zero for each fixed real or complex $x$. Both have infinite radius of convergence.