CalculusVolume2-OP-91

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important observation. Consider limn → ∞an + 1. Since
{an + 1} = {a2, a3, a4 ,…},
the only difference between the sequences {an + 1} and {an} is that {an + 1} omits the first term.
Since a finite number of terms does not affect the convergence of a sequence,
limn → ∞an + 1 = limn → ∞an = L.
Combining this fact with the equation
an + 1 = 4
n + 1an
and taking the limit of both sides of the equation
limn → ∞an + 1 = limn → ∞
4
n + 1an,
we can conclude that
L = 0 · L = 0.
b. Writing out the first several terms,
⎧
⎩
⎨2, 5
4, 41
40, 3281
3280,…
⎫
⎭
⎬.
we can conjecture that the sequence is decreasing and bounded below by 1. To show that the sequence
is bounded below by 1, we can show that
an
2 + 1
2an
≥ 1.
To show this, first rewrite
an
2 + 1
2an
= an
2 + 1
2an
.
Since a1 > 0 and a2 is defined as a sum of positive terms, a2 > 0. Similarly, all terms an > 0.
Therefore,
an
2 + 1
2an
≥ 1
if and only if
an
2 + 1 ≥ 2an.
Rewriting the inequality an
2 + 1 ≥ 2an as an
2 − 2an + 1 ≥ 0, and using the fact that
an
2 − 2an + 1 = (an − 1)2 ≥ 0
because the square of any real number is nonnegative, we can conclude that
an
2 + 1
2an
≥ 1.
Chapter 5 | Sequences and Series 443
5.6
To show that the sequence is decreasing, we must show that an + 1 ≤ an for all n ≥ 1. Since 1 ≤ an
2,
it follows that
an
2 + 1 ≤ 2an
2.
Dividing both sides by 2an, we obtain
an
2 + 1
2an
≤ an.
Using the definition of an + 1, we conclude that
an + 1 = an
2 + 1
2an
≤ an.
Since {an} is bounded below and decreasing, by the Monotone Convergence Theorem, it converges.
To find the limit, let L = limn → ∞an. Then using the recurrence relation and the fact that
limn → ∞an = limn → ∞an + 1, we have
limn → ∞an + 1 = limn → ∞
⎛
⎝
an
2 + 1
2an
⎞
⎠,
and therefore
L = L
2 + 1
2L.
Multiplying both sides of this equation by 2L, we arrive at the equation
2L2 = L2 + 1.
Solving this equation for L, we conclude that L2 = 1, which implies L = ±1. Since all the terms are
positive, the limit L = 1.
Consider the sequence {an} defined recursively such that a1 = 1, an = an − 1 /2. Use the Monotone
Convergence Theorem to show that this sequence converges and find its limit.
444 Chapter 5 | Sequences and Series
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Fibonacci Numbers
The Fibonacci numbers are defined recursively by the sequence {Fn} where F0 = 0, F1 = 1 and for n ≥ 2,
Fn = Fn − 1 + Fn − 2.
Here we look at properties of the Fibonacci numbers.
1. Write out the first twenty Fibonacci numbers.
2. Find a closed formula for the Fibonacci sequence by using the following steps.
a. Consider the recursively defined sequence {xn} where xo = c and xn + 1 = axn. Show that this
sequence can be described by the closed formula xn = can for all n ≥ 0.
b. Using the result from part a. as motivation, look for a solution of the equation
Fn = Fn − 1 + Fn − 2
of the form Fn = cλn. Determine what two values for λ will allow Fn to satisfy this equation.
c. Consider the two solutions from part b.: λ1 and λ2. Let Fn = c1 λ1
n + c2 λ2
n. Use the initial
conditions F0 and F1 to determine the values for the constants c1 and c2 and write the closed
formula Fn.
3. Use the answer in 2 c. to show that
limn → ∞
Fn + 1
Fn
= 1 + 5
2 .
The number ϕ = ⎛
⎝1 + 5⎞
⎠/2 is known as the golden ratio (Figure 5.8 and Figure 5.9).
Figure 5.8 The seeds in a sunflower exhibit spiral patterns
curving to the left and to the right. The number of spirals in each
direction is always a Fibonacci number—always. (credit:
modification of work by Esdras Calderan, Wikimedia
Commons)
Chapter 5 | Sequences and Series 445
Figure 5.9 The proportion of the golden ratio appears in many
famous examples of art and architecture. The ancient Greek
temple known as the Parthenon was designed with these
proportions, and the ratio appears again in many of the smaller
details. (credit: modification of work by TravelingOtter, Flickr)
446 Chapter 5 | Sequences and Series
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5.1 EXERCISES
Find the first six terms of each of the following sequences,
starting with n = 1.
1. an = 1 + (−1)n for n ≥ 1
2. an = n2 − 1 for n ≥ 1
3. a1 = 1 and an = an − 1 + n for n ≥ 2
4. a1 = 1, a2 = 1 and an + 2 = an + an + 1 for
n ≥ 1
5. Find an explicit formula for an where a1 = 1 and
an = an − 1 + n for n ≥ 2.
6. Find a formula an for the nth term of the arithmetic
sequence whose first term is a1 = 1 such that
an − 1 − an = 17 for n ≥ 1.
7. Find a formula an for the nth term of the arithmetic
sequence whose first term is a1 = −3 such that
an − 1 − an = 4 for n ≥ 1.
8. Find a formula an for the nth term of the geometric
sequence whose first term is a1 = 1 such that
an + 1
an
= 10 for n ≥ 1.
9. Find a formula an for the nth term of the geometric
sequence whose first term is a1 = 3 such that
an + 1
an
= 1/10 for n ≥ 1.
10. Find an explicit formula for the nth term of the
sequence whose first several terms are
{0, 3, 8, 15, 24, 35, 48, 63, 80, 99,…}. (Hint: First
add one to each term.)
11. Find an explicit formula for the nth term of the
sequence satisfying a1 = 0 and an = 2an − 1 + 1 for
n ≥ 2.
Find a formula for the general term an of each of the
following sequences.
12. {1, 0, −1, 0, 1, 0, −1, 0,…} (Hint: Find where
sinx takes these values)
13. {1, −1/3, 1/5, −1/7,…}
Find a function f (n) that identifies the nth term an of the
following recursively defined sequences, as an = f (n).
14. a1 = 1 and an + 1 = −an for n ≥ 1
15. a1 = 2 and an + 1 = 2an for n ≥ 1
16. a1 = 1 and an + 1 = (n + 1)an for n ≥ 1
17. a1 = 2 and an + 1 = (n + 1)an /2 for n ≥ 1
18. a1 = 1 and an + 1 = an /2n for n ≥ 1
Plot the first N terms of each sequence. State whether the
graphical evidence suggests that the sequence converges or
diverges.
19. [T] a1 = 1, a2 = 2, and for n ≥ 2,
an = 1
2(an − 1 + an − 2); N = 30
20. [T] a1 = 1, a2 = 2, a3 = 3 and for n ≥ 4,
an = 1
3(an − 1 + an − 2 + an − 3), N = 30
21. [T] a1 = 1, a2 = 2, and for n ≥ 3,
an = an − 1 an − 2; N = 30
22. [T] a1 = 1, a2 = 2, a3 = 3, and for n ≥ 4,
an = an − 1 an − 2 an − 3; N = 30
Suppose that limn → ∞an = 1, limn → ∞bn = −1, and
0 < −bn < an for all n. Evaluate each of the following
limits, or state that the limit does not exist, or state that
there is not enough information to determine whether the
limit exists.
23. limn → ∞3an − 4bn
24. limn → ∞
1
2bn − 1
2an
25. limn → ∞
an + bn
an − bn
26. limn → ∞
an − bn
an + bn
Find the limit of each of the following sequences, using
L’Hôpital’s rule when appropriate.
Chapter 5 | Sequences and Series 447