(a) Let f(x) = ln (1 + x). Then ,
and . At x = 0, f(0) = 0, f ′(0) = 1, f ′′(0) = −1,
f ′′′(0) = 2, f (4)(0) = −(3!), and f (5)(0) = 4!. So
(b) Using the Ratio Test, you know that the series converges when
, that is, when |x| < 1, or −1 < x < 1. Thus, the radius of
convergence is 1.
(c) ln .
(d) Since the series converges by the Alternating Series Test, the error in
the answer for (c) is less absolutely than .
From the equations for x and y,
dx = (1 − cos θ) dθ and dy = sin θ dθ.
(a) The slope at any point is given by , which here is . When
, the slope is .
(b) When and . The equation of the tangent
is .
