Question

The function \[ f(z) = (z - 1)^{-1} - 1 + (z - 1) - (z - 1)^2 + \cdots \] is the series expansion of

• A. \( \frac{-1}{z(z - 1)} \) for \( |z - 1| < 1 \)
• B. \( \frac{1}{z(z - 1)} \) for \( |z - 1| < 1 \)
• C. \( \frac{1}{(z - 1)^2} \) for \( |z - 1| < 1 \)
• D. \( \frac{-1}{(z - 1)} \) for \( |z - 1| < 1 \)

Hint:

The geometric series expansion is a standard method for representing functions in the form \( \frac{1}{z(z - 1)} \) when \( |z - 1| < 1 \).

Answer 1

The Correct Option is B

The given function \( f(z) \) is a series expansion. It is the geometric series expansion for the function \( \frac{1}{z(z - 1)} \) when \( |z - 1| < 1 \). The correct answer is option (B).

Step 1: The series corresponds to the expansion of \( \frac{1}{z(z - 1)} \), so option (B) is correct.

Step 2: The other options do not match the form of the series expansion.

Final Answer: (B) \( \frac{1}{z(z - 1)} \) for \( |z - 1| < 1 \)