Let f(x) = 0 anx be a formal power series centred at 0 with radius of...

 

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Let f(x) = 0 anx" be a formal power series centred at 0 with radius of convergence R > 0 (including R = 0). Show that as a function f(x) is even (respectively odd) if and only if an = 0 for all odd (respectively, even) n. Let f(x) = 0 anx" be a formal power series centred at 0 with radius of convergence R > 0 (including R = 0). Show that as a function f(x) is even (respectively odd) if and only if an = 0 for all odd (respectively, even) n.