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Radius of convergence of binomial series for fixed complex x

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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.

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Solution Summary

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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