Complex analysis examples discussion 03

[03.1] For a bounded sequence of complex numbers c n , prove that ∞ n=0 c n z n z n + 1 converges to a holomorphic function on |z| < 1. Each summand is holomorphic on |z| < 1, because of the quotient rule, and that the numerator and denominator are polynomials, hence holomorphic. To prove that the sum n f n of a sequence of holomorphic functions on |z| < 1 is itself holomorphic, it suffices to prove that the convergence is uniform on compacts. The compact subsets of the open disk are all contained in compact disks |z| ≤ r for r < 1, so it suffices to consider just those sets |z| ≤ r.