Math 6420 • PDE

Function \(f(x)\) (see below) is expanded in powers of \(x-x_0\), i.e. in the Taylor or Laurent series with center \(x_0\) (\(x\) is a real variable, \(x_0\) is a real number). Find the radius of convergence of this series. \[ \begin {array}{rcll} \text {(a)}\quad f(x)&=&\dfrac {\sin x^2}{(x-3)^2(x^2+1)},& x_0=3,\\[1.4ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x}{x},& x_0=5,\\[1.4ex] \text {(c)}\quad f(x)&=&\dfrac {\sin x}{\sin x+3},& x_0=8. \end {array} \]