The complex series expansion of a function f(z) about a point z=a is given by the formula:

f(z) = Σ (n=0 to ∞) [c_n * (z-a)^n]

where c_n is the nth coefficient of the series expansion. In this case, we want to find the radius of convergence of the complex series expansion of 1/(3-2z) about z=2.

1. Finding the Coefficients:
To find the coefficients c_n, we can use the formula for the nth derivative of f(z) at z=a:

c_n = f^n(a) / n!

In this case, the function f(z) = 1/(3-2z), so we need to find the nth derivative of f(z) and evaluate it at z=2.

2. Derivatives of f(z):
Let's find the first few derivatives of f(z):

f'(z) = 2/(3-2z)^2
f''(z) = 8/(3-2z)^3
f'''(z) = 48/(3-2z)^4

We can observe a pattern in the derivatives:

f^n(z) = (-2)^n * n! / (3-2z)^(n+1)

3. Evaluating the Coefficients:
Now, let's evaluate the nth derivative at z=2 to find the coefficients c_n:

c_n = f^n(2) / n!
= (-2)^n * n! / (3-2*2)^(n+1)
= (-2)^n * n! / (3-4)^(n+1)
= (-2)^n * n! / (-1)^(n+1)

4. Simplifying the Coefficients:
Since (-1)^(n+1) equals 1 for odd values of n and -1 for even values of n, we can simplify the coefficients:

c_n = (-2)^n * n! / (-1)^(n+1)
= (-2)^n * n! * (-1)^n
= (-1)^n * 2^n * n!

5. Convergence Criteria:
The radius of convergence, denoted as R, can be determined using the ratio test:

R = lim (n→∞) |c_n / c_(n+1)|

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6. Applying the Ratio Test:
Let's apply the ratio test to find the radius of convergence:

R = lim (n→∞) |(-1)^n * 2^n * n! / (-1)^(n+1) * 2^(n+1) * (n+1)!|
= lim (n→∞) |(-1)^n * 2^n * n! / (-1)^(n+1) * 2^(n+1) * (n+1) * n!|
= lim (n→∞) |(-1)^n * 2^n / (-1)^(n+1) * 2^(n+1) * (n+1)