# Radius of convergence of binomial series for fixed complex x

The generalized binomial coefficient for z in C (complex) and k = 0, 1, 2, ... is

1 if k = 0 and

( k )

( z ) = [z(z-1) ... (z - k +1)] / k!

if k >=1. For a fixed value of s in C we define the binomial series

B_s*(z) = sum of n=0, ^infinity, ( s )

( n ) z^n

Find the radius of convergence of B_s.

See the formatted question attached.

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In this solution, we show that the radius of convergence of the binomial series is 1 for all complex values of its argument.

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