Suppose a power series T(x)=∑n=0∞an(x)n has a finite, non-zero radius of convergence R. Define the new series
U(x)=n=0∑∞(n+1)an+1xn,
which is obtained by differentiating T(x) term-by-term.
Address the following prompts.
a. Explain, without direct computation of limits, why U(x) must have the same radius of convergence R as T(x). Your explanation should reference relevant properties of power series.
b. Provide a specific example of a series T(x) with finite radius R and verify, through explicit calculation, that both T and U indeed share this radius.
c. Discuss why equality of radii does not guarantee identical behavior at corresponding endpoints, illustrating your discussion with the example chosen in part (b).