CalculusVolume2-SASG-05-01
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 5
Sequences and Series
5.1 Sequences
Section Exercises
Find the first six terms of each of the following sequences, starting with 1.
for Answer: if is odd and if is even
3.
and for Answer: 5.
Find an explicit formula for where and for Answer: 7.
Find a formula for the term of the arithmetic sequence whose first term is
such that for Answer: 9.
Find a formula for the term of the geometric sequence whose first term is such that for Answer: 11.
Find an explicit formula for the term of the sequence satisfying and
for Answer: Find a formula for the general term of each of the following sequences.
13.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: Find a function that identifies the term of the following recursively defined sequences, as 15.
and for Answer: 17.
and for Answer: Plot the first terms of each sequence. State whether the graphical evidence suggests that
the sequence converges or diverges.
19.
[T]
and for Answer: Terms oscillate above and below and appear to converge to
OpenStax Calculus Volume 2
Student Answer and Solution Guide
21.
[T]
and for Answer: Terms oscillate above and below and appear to converge to a limit.
Suppose that and for all 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.
Answer: 25.
Answer: Find the limit of each of the following sequences, using L’Hôpital’s rule when appropriate. 27.
Answer:
29.
Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
For each of the following sequences, whose terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing.
31.
Answer: bounded, decreasing for 33.
Answer: bounded, not monotone
35.
Answer: bounded, decreasing
37.
Answer: not monotone, not bounded
39.
Determine whether the sequence defined as follows has a limit. If it does, find the limit
Answer: is decreasing and bounded below by The limit must satisfy so independent of the initial value.
Use the Squeeze Theorem to find the limit of each of the following sequences.
41.
Answer: 43.
Answer: :
and so
OpenStax Calculus Volume 2
Student Answer and Solution Guide
For the following sequences, plot the first terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges.
45.
[T]
Answer: Graph oscillates and suggests no limit.
Determine the limit of the sequence or show that the sequence diverges. If it converges, find
its limit.
47.
Answer: and so 49.
Answer: Since one has as 51.
Answer: and as so as 53.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: In particular,
so as Newton’s method seeks to approximate a solution that starts with an initial approximation and successively defines a sequence For the given choice of and write out the formula for If the sequence appears to converge, give an exact formula for the solution then identify the limit accurate to four decimal places and the smallest such that agrees with up to four decimal places.
55.
[T]
Answer: 57.
[T]
,
Answer:
59.
[
T]
A lake initially contains fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by each month. However, factoring in all causes, fish are lost each month.
a.
Explain why the fish population after months is modeled by with
b.
How many fish will be in the pond after one year?
Answer: a. Without losses, the population would obey The subtraction of accounts for fish losses. b. After months, we have 61.
[
T]
A student takes out a college loan of at an annual percentage rate of
compounded monthly. a.
If the student makes payments of per month, how much does the student owe after
months? b.
After how many months will the loan be paid off?
Answer: a. The student owes after months. b. The loan will be paid in full after months or eleven and a half years
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
63.
[T]
The binary representation of a number between and can be defined as follows. Let if and if Let Let
if and if Let and in general,
and if and if Find the binary expansion of Answer:
so the pattern repeats, and
OpenStax Calculus Volume 2
Student Answer and Solution Guide
For the following two exercises, assume that you have access to a computer program or Internet source that can generate a list of zeros and ones of any desired length. Pseudorandom number generators (PRNGs) play an important role in simulating random noise in physical systems by creating sequences of zeros and ones that appear like the result
of flipping a coin repeatedly. One of the simplest types of PRNGs recursively defines a random-looking sequence of integers by fixing two special integers and
and letting be the remainder after dividing into then creates a bit sequence of zeros and ones whose term is equal to one if is odd and equal to zero
if is even. If the bits are pseudorandom, then the behavior of their average
should be similar to behavior of averages of truly randomly generated
bits.
65.
[
T]
Starting with and using ten different starting values
of compute sequences of bits up to and compare their averages to ten such sequences generated by a random bit generator.
Answer: For the starting values the corresponding bit averages calculated by the method indicated are and
Here is an example of ten corresponding averages of strings of bits generated by a random number generator: There is no real pattern in either type of average. The random-number-generated averages range between and a range of
whereas the calculated PRNG bit averages range between and a range of
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Student Project
Fibonacci Numbers
1.
Write out the first twenty Fibonacci numbers.
Answer: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181
3.
Use the answer in 2 c. to show that Answer:
We notice that the denominator is a geometric series with ratio Therefore, the denominator can be rewritten as Therefore, we can write is a geometric series with ratio
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
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