(C)
Since the series converges when , that is, when , the radius of
convergence is .
(C) This polynomial is associated with the binomial series (1 + x)p.
Verify that f(0) = 1, f ′(0) = p, f ′′(0) = p(p − 1), f ′′′(0) = p(p − 1)(p − 2).
(A) The fastest way to find the series for ln(1 + 2x) about x = 0 is to
substitute 2x for x in the series
(D) . The series therefore converges if . which
is less than 1 if 2 < x. If x < 0, , which is less than 1 if−2 > x. Now
for the endpoints:
x = 2 yields 1 + 1 + 1 + 1 + . . . , which diverges;
x = −2 yields −1 + 1 − 1 + 1 − . . . , which diverges.
The answer is |x| > 2.
(C) The function and its first three derivatives at are sin ; cos
; −sin ; and −cos . P 3 (x) is choice C.
