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tylerc1991
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Homework Statement
Suppose that: sum [a_n (n-1)^n] is the Talyor series representation of tanh(z) at the point z = 1. What is the largest subset of the complex plane such that this series converges?
Note: 'sum' represents the sum from n=0 to infinity
Homework Equations
tanh(z) = sinh(z)/cosh(z) ; cosh(z) = 0 for z = pi*i/2
B(C,R) is the open ball centered at C and with radius R
d(x,y) is the distance between x and y
The Attempt at a Solution
So the series is centered at z = 1 and the first place that tanh(z) is not analytic (expanding from z = 1) is at the point z = pi*i/2. so I can create an open ball of radius d(1,pi*i/2) and this is the largest subset of convergence.
d(1,pi*i/2) = (1/2)sqrt(4 + pi^2)
so the largest subset in which the Taylor series converges is B(1,(1/2)sqrt(4 + pi^2))