What is true for a power series with the general term aix^i?

A power series is expressed in the form of the general term ai x^i, where the sum of these terms can define a function within a specific interval of convergence. The correctness of the chosen statement stems from the properties of power series regarding their convergence and the operations that can be performed on them.

Within the interval of convergence, power series have the flexibility to be manipulated in various ways. Specifically, they can be differentiated and integrated term-by-term. This means that if you have a power series that converges, you can obtain a new power series by differentiating it or integrating it, and this new series will also converge within the same interval (or a suitably modified interval depending on the operation).

Moreover, power series can also be added and subtracted term-by-term as long as the series involved converge in the same interval. This is a powerful feature that allows the construction of new functions from existing ones.

The other statements do not hold true in the same way. For instance, while a power series might converge at certain endpoints, it is not restricted only to that condition. In addition, power series can indeed be differentiated multiple times, depending on their radius of convergence, allowing for repeated applications of differentiation. Lastly, although power series can represent many