MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
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.
(1) 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.
(2) 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.
(3) Discuss why equality of radii does not guarantee identical behavior at corresponding endpoints, illustrating your discussion with the example chosen in part (b).