identity theorem of power series

If the radii of convergence (http://planetmath.org/RadiusOfConvergence) of the power series  ${\sum }_{n=0}^{\mathrm{\infty }}{a}_n{(z-z_0)}^n$  and  ${\sum }_{n=0}^{\mathrm{\infty }}{b}_n{(z-z_0)}^n$  are positive and the sums of the series are equal in infinitely many points which have $z_0$ as an accumulation point, then the both series are identical, i.e.  $a_n=b_n$  for each  $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$