This question is a true/false problem about the properties of a complex function $f(z)$ within a certain region in the complex plane, particularly about its differentiability and the existence of derivatives of any order. The statement in the image seems to be claiming that $f(z)$, which is the sum of a power series, is differentiable of any order within a disk of convergence, suggesting that the function is analytic.

Explanation:

1. Analytic Function: If a function $f(z)$ is represented by a convergent power series within a disk, it is analytic in that region. An analytic function is infinitely differentiable and its derivatives of all orders exist within that region.

2. True or False: Given the properties of analytic functions, the statement in option A is correct (true), assuming the function is analytic within the disk of convergence.

Conclusion:

The answer is therefore True (A), indicating that the function has derivatives of any order within the disk.

Would you like a detailed explanation of analytic functions, their derivatives, or how they relate to power series?

Further Questions:

1. What are the properties of analytic functions in the complex plane?
2. How does the radius of convergence affect the differentiability of a power series?
3. What is the difference between analytic and meromorphic functions?
4. How can we determine the radius of convergence for a given power series?
5. Why does having derivatives of all orders imply that a function is analytic?

Tip: For functions represented by power series, if the series converges within a certain radius, the function is not only continuous but also differentiable infinitely often within that radius.