Question

The radius of convergence of Taylor's series expansion of \( f(z) = \frac{1}{(z - 1)^2} \) in powers of \( (z - 1) \) is ...

• A. 1
• B. 0
• C. \( \infty \)
• D. -1

Hint:

The radius of convergence of a Taylor series is the distance from the center of expansion to the nearest singularity of the function.

Answer 1

The Correct Option is C

The radius of convergence of the Taylor series for the function \( f(z) = \frac{1}{(z - 1)^2} \) around \( z = 1 \) is determined by the distance from the expansion point to the nearest singularity. Since the function has a singularity at \( z = 1 \), the radius of convergence is infinite.