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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{5(1-1.01)n}
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). -0.027 = 0.027027..
Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{1.00001n}
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.12 = 0.121212...
Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{2n + 13-n}
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 1.25 = 1.252525.
Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{(-2.5)n}
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.456 = 0.456456456...
Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.{100(-0.003)n}
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 1.0039 = = 1.00393939...
Find the limit of the following sequences or state that they diverge. cos n n
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.00952 = 0.00952952..
Find the limit of the following sequences or state that they diverge. sin 6n 5n
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 5.1283 = 5.12838383.
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim Sn n→∞ Σ k=1 1 k + 1 1 k + 2
Find the limit of the following sequences or state that they diverge. sin n 2n
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim Sn n→∞ n k=1 1 k + 2 1 k + 3
Find the limit of the following sequences or state that they diverge. cos (nπ/2) COS Vn
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim Sn n→∞ 1 Σ Ξ (k + 6)(k + 7) k=1
Find the limit of the following sequences or state that they diverge. 2 tan¯¹ n n³ + 4 3
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim Sn n→∞ 1 k=0 (3k + 1)(3k + 4) A
Find the limit of the following sequences or state that they diverge. n sin³ (nπ/2) n+ (NT/2)}
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim Sn n→∞ Σ k=3 4 (4k – 3)(4k + 1)
Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume also that aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.a. Find a recurrence relation for
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim Sn n→∞ Σ k=3 2 (2k – 1)(2k + 1)
Marie takes out a $20,000 loan for a new car. The loan has an annual interest rate of 6% or, equivalently, a monthly interest rate of 0.5%. Each month, the bank adds interest to the loan balance (the interest is always 0.5% of the current balance), and then Marie makes a $200 payment to reduce the
James begins a savings plan in which he deposits $100 at the beginning of each month into an account that earns 9% interest annually or, equivalently, 0.75% per month. To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits $100.
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate to obtain the value of the series or state that the series diverges. lim Sn n→∞ Σ(√k + 1 √k) k=1
A tank is filled with 100 L of a 40% alcohol solution (by volume). You repeatedly perform the following operation: Remove 2 L of the solution from the tank and replace them with 2 L of 10% alcohol solution.a. Let Cn be the concentration of the solution in the tank after the nth replacement, where
Assume that (Exercises 65 and 66) and that the terms of this series may be rearranged without changing the value of the series. Determine the sum of the reciprocals of the squares of the odd positive integers. Data from Exercise 65 The Riemann zeta function is the subject of extensive research and
Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. en/10 2n
Suppose that you open a savings account by depositing $100. The account earns interest at an annual rate of 3% per year (0.25% per month). At the end of each month, you earn interest on the current balance, and then you deposit $100. Let Bn be the balance at the beginning of the nth month, where B0
Use the formal definition of the limit of a sequence to prove the following limits. 1 lim n→∞ n = 0
Use the formal definition of the limit of a sequence to prove the following limits. 1 lim n→∞ n 2 = 0
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.6 = 0.666...
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an In n 1.1 S
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 0.3 = 0.333...
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility.an = e-n cos n
Evaluate each geometric series or state that it diverges. Σ 3 k=1 8 3k
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an || e -n 2 sin(e)
Evaluate each geometric series or state that it diverges. ∞o Σ(-0.15)* k=2
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility.an = 1 + cos 1/n
Evaluate each geometric series or state that it diverges. Σ(-e) * k=1
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an 3n 3h+4"
Evaluate each geometric series or state that it diverges. 3Σ (-π) * k=0
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility. an sin (nπ/3) Vn
Evaluate each geometric series or state that it diverges. Σ k=1 k 2 3
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility.an = (-1)n n/n + 1
Evaluate each geometric series or state that it diverges. Σ k=0 k 9 10/
Find the limit of the following sequences or determine that the limit does not exist. Verify your result with a graphing utility.an = sin nπ/2
Evaluate each geometric series or state that it diverges. ∞o k=2 3 100 8 3k
Evaluate each geometric series or state that it diverges. k=0 4 k 53-k
Evaluate each geometric series or state that it diverges. 1 3 + 16 64 + 9 + 256 27 1024 +
Evaluate each geometric series or state that it diverges. 1 + TT บ 2 TT + e3 3 T +
Evaluate each geometric series or state that it diverges. k=0 4 3 -k
Evaluate each geometric series or state that it diverges. 8 Σ Sk k=4
Evaluate each geometric series or state that it diverges. k=3 3.4k 7k
Evaluate each geometric series or state that it diverges. 8 Σ2-3 k=1
Evaluate each geometric series or state that it diverges. 8 Σ m = 2 5 2m
Evaluate each geometric series or state that it diverges. 8 Σε k=1 −2k
Evaluate each geometric series or state that it diverges. 1 + 1 TT + - TT - _3 TT +
Evaluate each geometric series or state that it diverges.1 + 1.01 + 1.012 + 1.013 + . . .
Evaluate each geometric series or state that it diverges. 1 + 2 7 + 2² 7² + 73
Evaluate each geometric series or state that it diverges. 20.94 k=0
Evaluate each geometric series or state that it diverges. 8 Σ k=0 3 5 k
Evaluate each geometric series or state that it diverges. 8 Σ k=0 4 k
Evaluate each geometric sum. 1 4 + -19 12 + |- 1 36 + 1 108 + 1 2916
Evaluate each geometric sum. 1 3 + 5 25 + 9 125 + 243 15,625
Evaluate each geometric sum. 1 + 2/3 + 4 8 + 9 27
Evaluate each geometric sum. 20 k Σ(-1)* k=0
Evaluate each geometric sum. Σ(;) k=1 k
Evaluate each geometric sum. 6 Σπ k=0 k
Evaluate each geometric sum. 20 W k=0 26 2 5
Evaluate each geometric sum. 8 Σ34 k=0
Evaluate each geometric sum. 5 Σ(-2.5)* k=1
What is the condition for convergence of the geometric series? 8 Σαγκ k=0
Evaluate each geometric sum. 12 Σ2* k=4
Evaluate each geometric sum. 10 Σ k=0 4 k
Evaluate each geometric sum. Σ k=0 3 4 k
Does a geometric series always have a finite value?
Does a geometric sum always have a finite value?
What is meant by the ratio of a geometric series?
What is the difference between a geometric sum and a geometric series?
What is the defining characteristic of a geometric series? Give an example.
Can the Integral Test be used to determine whether a series diverges?
Is it true that if the terms of a series of positive terms decrease to zero, then the series converges? Explain using an example.
If we know thatthen what can we say about lim az = 1, k- Σ ακ k=1
a. Consider the number 0.555555 . . ., which can be viewed as the series Evaluate the geometric series to obtain a rational value of 0.555555 . .b. Consider the number 0.54545454 . . ., which can be represented by the series Evaluate the geometric series to obtain a rational value of the
The fractal called the snowflake island (or Koch island) is constructed as follows: Let I0 be an equilateral triangle with sides of length 1. The figure I1 is obtained by replacing the middle third of each side of I0 with a new outward equilateral triangle with sides of length 1/3 (see figure). The
Imagine that the government of a small community decides to give a total of $W, distributed equally, to all its citizens. Suppose that each month each citizen saves a fraction p of his or her new wealth and spends the remaining 1 - p in the community. Assume no money leaves or enters the community,
Suppose a rubber ball, when dropped from a given height, returns to a fraction p of that height. In the absence of air resistance, a ball dropped from a height h requires √2h/g seconds to fall to the ground, where g ≈ 9.8 m/s2 is the acceleration due to gravity. The time taken to bounce up to a
An insulated window consists of two parallel panes of glass with a small spacing between them. Suppose that each pane reflects a fraction p of the incoming light and transmits the remaining light. Considering all reflections of light between the panes, what fraction of the incoming light is
In 1978, in an effort to reduce population growth, China instituted a policy that allows only one child per family. One unintended consequence has been that, because of a cultural bias toward sons, China now has many more young boys than girls. To solve this problem, some people have suggested
Suppose that you take 200 mg of an antibiotic every 6 hr. The half-life of the drug is 6 hr (the time it takes for half of the drug to be eliminated from your blood). Use infinite series to find the long-term (steady-state) amount of antibiotic in your blood.
A fishery manager knows that her fish population naturally increases at a rate of 1.5% per month. At the end of each month, 120 fish are harvested. Let Fn be the fish population after the nth month, where F0 = 4000 fish. Assume that this process continues indefinitely. Use infinite series to find
Suppose you borrow $20,000 for a new car at a monthly interest rate of 0.75%. If you make payments of $600/month, after how many months will the loan balance be zero? Estimate the answer by graphing the sequence of loan balances and then obtain an exact answer using infinite series.
Suppose you take out a home mortgage for $180,000 at a monthly interest rate of 0.5%. If you make payments of $1000/month, after how many months will the loan balance be zero? Estimate the answer by graphing the sequence of loan balances and then obtain an exact answer using infinite series.
a. Evaluate the seriesb. For what values of a does the seriesconverge, and in those cases, what is its value? 3k (3*+1 – 1)(3* – 1) 1) |k=1 Σ ak k=1 (ak+1 1)(a² 8.
The Greeks solved several calculus problems almost 2000 years before the discovery of calculus. One example is Archimedes’ calculation of the area of the region R bounded by a segment of a parabola, which he did using the “method of exhaustion.” As shown in the figure, the idea was to fill R
The Greek philosopher Zeno of Elea (who lived about 450 b.c.) invented many paradoxes, the most famous of which tells of a race between the swift warrior Achilles and a tortoise. Zeno argued The slower when running will never be overtaken by the quicker; for that which is pursuing must first
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