A student applies a convergence test to a power series centered at the origin and finds that the series converges whenever ∣x∣<4.
(a) Explain the meaning of the statement ∣x∣<4 in the context of power-series convergence. What information does it provide and what information does it omit?
(b) Give a detailed reason why the same test used by the student provides no conclusion about convergence at the points x=±4.
(c) Construct an explicit power series centered at 0 that has radius 4 and converges at x=4. Justify the convergence at that point.
(d) Construct a different power series (also centered at 0 with radius 4) that diverges at x=4. Justify the divergence at that point.