Math 6420 • PDE

The following function \(f(x)\) [\(x\) is a real variable] can be represented by a series (Taylor or Laurent) in powers of \(x\). Find the radius of convergence of the series in each case \[ \begin {array}{rcl} \text {(a)}\quad f(x)&=&\left (\dfrac {\sin x}{x}\right )^2,\\[1.4ex] \text {(b)}\quad f(x)&=&\dfrac {\sin x}{x^2},\\[1.4ex] \text {(c)}\quad f(x)&=&\left (\dfrac {x}{\sin x}\right )^2,\\[1.4ex] \text {(d)}\quad f(x)&=&\dfrac {x^2}{\sin x},\\[1.4ex] \text {(e)}\quad f(x)&=&\dfrac {\sin x}{x^2+1}. \end {array} \] [You do not need to find the series themselves.]