From Thomas, 9th edition, Page 700, 42-50, 58-76 evens. Note that other correct reasons exist for these answers.
42) åan has limn®¥|an+1/an| = 0 so the series is absolutely convergent for all values of x.
44) åan has limn®¥|an+1/an| = |2x+1|/2. |2x+1|/2 < 1 will imply the series converges absolutely for -3/2 < x <1/2. The series diverges at x = -3/2 and x = 1/2 as limn®¥|an| = 1/2.
46) åan has limn®¥|an+1/an| = |x| so the series will converge absolutely for -1 < x < 1. At x = -1, the series satisfies the alternating series test, so we have conditional convergence. The series diverges at x = 1.
48) åan has limn®¥|an+1/an| = |x-1|2. |x-1|<1 implies that the series converges absolutely for 0 < x < 2. At the endpoints, the series satisfies the alternating series test, so it is conditionally convergent at x = 0 and x = 2.
50) åan has limn®¥|an+1/an| = |x| so the series is absolutely convergent for all values of -1 < x < 1. The series diverges at the endpoints.
58) 1/(1+x3) = 1/(1-(-x3)) = 1 - x3 + x9 - ¼ = ån = 0¥(-1)n x3n.
60) sin(2x/3) = 2x/3 - (2x/3)3/3! + (2x/3)5/5! - ¼ = ån = 0¥(-1)n (2x/3)2n+1/(2n+1)!
62) cos[Ö5x] = 1 - 5x/2! + 25x2/4! - ¼ = ån = 0¥(-1)n (5x)n/(2n)!
64) e-x2 = 1-x2+x4/2! - ¼ = ån = 0¥ (-1)n x2n/n!
66) f(x) = -1 + 1(x-2)/1! -2(x-2)2/2! + 6(x-2)3/3! - ¼
68) f(x) = 1/a - 1/a2(x-a)/1! +2/a3(x-a)2/2! - 6/a4(x-a)3/3! + ¼
70) y = -3-3x-3/2 x2 -1/2 x3 -1/8 x4 - ¼
72) y = x - 1/2 x2 + 1/6 x3 - 1/24 x4 + ¼
74) y = 1/2 x2 - 1/6 x3 + 1/24 x4 + ¼
76) y = 2 + 2x + 1/2 x2 + 1/6 x3 + 1/24 x4 + ¼
From Thomas, 9th edition, page 858, 66-76 and 84-110, evens.
66) A sphere of radius 1 centered at (0,1,0) that intercepts the points (0,0,0), (0,2,0), (0,1,1), (0,1,-1), (1,1,0) and (-1,1,0). The trace in the xy plane is the circle x2+(y-1)2 = 1, in the xz plane is the circle x2+z2 = 1 and in the yz plane is the circle (y-1)2+z2 = 1.
68) An ellipsoid centered at the origin that intercepts the coordinate axes at the points (±1, 0,0), (0,±2, 0) and (0,0,±3). The trace in the xy plane is the ellipse x2+y2/4 = 1, in the xz plane is the ellipse x2+z2/9 = 1 and in the yz plane is the ellipse y2/4+z2/9 = 1.
70) A circular paraboloid with vertex at the origin that sits on the negative y axis. The trace in the xy plane is the parabola y = -x2 and in the yz plane is the parabola y = -z2.
72) A circular cone that is centered at the origin that sits on the y axis. The trace in the xy plane are the lines y = ±x and in the yz plane are the lines y = ±z.
74) A hyperboloid of one sheet that is centered at the origin with traces of the ellipse y2+z2/4 = 1 in the yz plane, the hyperbola y2-x2 = 1 in the xy plane and the hyperbola z2/4-x2 = 1 in the xz plane.
76) A hyperboloid of two sheets that intercepts the z axis at the points (0,0,±1) with the traces of the hyperbola z2-x2 = 1 in the xz plane and the hyperbola z2-y2 = 1 in the yz plane.
84) A circular cylinder that is parallel to the z axis that has a trace in the xy plane of a circle of radius 1/2 centered at (0,1/2,0).
86) A cylinder that is parallel to the z axis that has a trace in the xy plane of a 4 petaled rose where each of the petals intercepts the coordinate axes at (±1, 0,0) and (0,±1, 0).
88) A plane that is parallel to the z axis that intercepts the xy plane on the line y = x.
90) The unit circle.
92) A hollow sphere centered at the origin that is hollow from the origin to a distance of 1, and then solid from a distance of 1 to a distance of 2.
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