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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

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
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
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
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
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
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;
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
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
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
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
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
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
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
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
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
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
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
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.
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,
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
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,
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
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
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
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

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