James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

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FIGURE for problem 12
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11. Find the interval of convergence of oǹ−1 n3 x n and find its sum.
12. Suppose you have a large supply of books, all the same size, and you stack them at the edge
of a table, with each book extending farther beyond the edge of the table than the one
beneath it. Show that it is possible to do this so that the top book extends entirely beyond
the table. In fact, show that the top book can extend any distance at all beyond the edge of
the table if the stack is high enough. Use the following method of stacking: The top book
extends half its length beyond the second book. The second book extends a quarter of its
length beyond the third. The third extends one-sixth of its length beyond the fourth, and so
on. (Try it yourself with a deck of cards.) Consider centers of mass.
13. Find the sum of the series ò lnS1 2 1
n−2 n 2D.
14. If p . 1, evaluate the expression
1 1 1 2 p 1 1 3 p 1 1 4 p 1 ∙ ∙ ∙
1 2 1 2 p 1 1 3 p 2 1 4 p 1 ∙ ∙ ∙
FIGURE for problem 15
P¡ P∞
P
P˜
Pˆ
Pß
P¡¸
P¢ P
P£
FIGURE for problem 18
15. Suppose that circles of equal diameter are packed tightly in n rows inside an equilateral triangle.
(The figure illustrates the case n − 4.) If A is the area of the triangle and A n is the
total area occupied by the n rows of circles, show that
A n
lim
n l ` A −
2s3
16. A sequence ha nj is defined recursively by the equations
a 0 − a 1 − 1
Find the sum of the series
nsn 2 1da n − sn 2 1dsn 2 2da n21 2 sn 2 3da n22
oǹ−0 an.
17. If the curve y − e 2xy10 sin x, x > 0, is rotated about the x-axis, the resulting solid looks like
an infinite decreasing string of beads.
(a) Find the exact volume of the nth bead. (Use either a table of integrals or a computer
algebra system.)
(b) Find the total volume of the beads.
18. Starting with the vertices P 1s0, 1d, P 2s1, 1d, P 3s1, 0d, P 4s0, 0d of a square, we construct
further points as shown in the figure: P 5 is the midpoint of P 1P 2, P 6 is the midpoint of
P 2P 3, P 7 is the midpoint of P 3P 4, and so on. The polygonal spiral path P 1P 2P 3P 4P 5P 6P 7 . . .
approaches a point P inside the square.
(a) If the coordinates of P n are sx n, y nd, show that 1 2 xn 1 xn11 1 xn12 1 xn13 − 2 and find a
similar equation for the y-coordinates.
(b) Find the coordinates of P.
s21d
19. Find the sum of the series ò
n
n−1 s2n 1 1d3 . n
20. Carry out the following steps to show that
1
1 ? 2 1 1
3 ? 4 1 1
5 ? 6 1 1
7 ? 8 1 ∙ ∙ ∙ − ln 2
(a) Use the formula for the sum of a finite geometric series (11.2.3) to get an expression for
1 2 x 1 x 2 2 x 3 1 ∙ ∙ ∙ 1 x 2n22 2 x 2n21
789
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