Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 3 / a / Solution

The function $f$ is real analytic at $y$ if there is a neighborhood of $y$ on which its multivariable Taylor seriesconverges to $f(x)$. Here $\alpha$ is a multi-index; equivalently, $f$ agrees locally with a convergent real power series centred at $y$.