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

(b) Integrate the result of part (a) from 0 to 1 to get an expression for
1 2 1 2 1 1 3 2 1 4 1 ∙ ∙ ∙ 1 1
2n 2 1 2 1
2n
as an integral.
(c) Deduce from part (b) that
1
Z
1 ? 2 1 1
3 ? 4 1 1
5 ? 6 1 ∙ ∙ ∙ 1 1
s2n 2 1ds2nd 2 y 1
0
dx
1 1 x
Z , y 1
0
x 2n dx
(d) Use part (c) to show that the sum of the given series is ln 2.
21. Find all the solutions of the equation
1 1 x 2! 1 x 2
4! 1 x 3
6! 1 x 4
8! 1 ∙ ∙ ∙ − 0
[Hint: Consider the cases x > 0 and x , 0 separately.]
1
1
1
1
22. Right-angled triangles are constructed as in the figure. Each triangle has height 1 and its
base is the hypotenuse of the preceding triangle. Show that this sequence of triangles makes
indefinitely many turns around P by showing that g n is a divergent series.
¨£
¨
¨¡
P
FIGURE for problem 22
1
1
23. Consider the series whose terms are the reciprocals of the positive integers that can be written
in base 10 notation without using the digit 0. Show that this series is convergent and the
sum is less than 90.
24. (a) Show that the Maclaurin series of the function
f sxd −
x
1 2 x 2 x is ò f nx n
2
n−1
where f n is the nth Fibonacci number, that is, f 1 − 1, f 2 − 1, and f n − f n21 1 f n22
for n > 3. [Hint: Write xys1 2 x 2 x 2 d − c 0 1 c 1x 1 c 2x 2 1 . . . and multiply both
sides of this equation by 1 2 x 2 x 2 .]
(b) By writing f sxd as a sum of partial fractions and thereby obtaining the Maclaurin series
in a different way, find an explicit formula for the nth Fibonacci number.
25. Let u − 1 1 x 3
v − x 1 x 4
3! 1 x 6
6! 1 x 9
w − x 2
2! 1 x 5
5! 1 x 8
Show that u 3 1 v 3 1 w 3 2 3uvw − 1.
9! 1 ∙ ∙ ∙
4! 1 x 7
7! 1 x 10
10! 1 ∙ ∙ ∙
8! 1 ∙ ∙ ∙
26. Prove that if n . 1, the nth partial sum of the harmonic series is not an integer.
Hint: Let 2 k be the largest power of 2 that is less than or equal to n and let M be the product
of all odd integers that are less than or equal to n. Suppose that s n − m, an integer. Then
M2 k s n − M2 k m. The right side of this equation is even. Prove that the left side is odd by
showing that each of its terms is an even integer, except for the last one.
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