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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

Find two divergent series Σ an and Σ bn such that Σ(an + bn) converges.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. n=1 L, then a₁ = L + ao. n=0
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If |r| < 1, then arn n=1 a 1-r
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Every decimal with a repeating pattern of digits is a rational number.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.0.75 = 0.749999 . . . .
Prove Theorem 9.5 for a nonincreasing sequence.Data from in Theorem 9.5 THEOREM 9.5 Bounded Monotonic Sequences If a sequence {a} is bounded and monotonic, then it converges.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If lim an = n-x 0, then a, converges. n=1
In Exercises consider making monthly deposits of P dollars in a savings account at an annual interest rate r. Use the results of Exercise 84 to find the balance A after t years when the interest is compounded (a) Monthly (b) ContinuouslyData from in Exercise 84When an employee receives a
Find, with proof, a formula for Tn, of the form T₁ = An +Bn where {An} and {Bn} are well-known sequences.Let T0 = 2, T1 = 3, T₂ = 6, and for n ≥ 3,T₁ = (n + 4)Tn−1 − 4nTn-2 + (4n − 8)Tn-3.The first few terms are2, 3, 6, 14, 40, 152, 784, 5168, 40,576
Let {xn}, n ≥ 0, be a sequence of nonzero real numberssuch that x2n - xn-1 Xn+1 = 1 for n = 1, 2, 3, . . . . Provethere exists a real number a such that Xn+1 = axn - xn-1 for all n ≥ 1.
In Exercises consider making monthly deposits of P dollars in a savings account at an annual interest rate r. Use the results of Exercise 84 to find the balance A after t years when the interest is compounded (a) Monthly (b) ContinuouslyData from in Exercise 84When an employee receives a
Prove, using the definition of the limit of a sequence, that lim r = 0 for -1 < r < 1. n n18
In Exercises consider making monthly deposits of P dollars in a savings account at an annual interest rate r. Use the results of Exercise 84 to find the balance A after t years when the interest is compounded (a) Monthly (b) ContinuouslyData from in Exercise 84When an employee receives a
Prove, using the definition of the limit of a sequence, that 1 lim = 0. n³ 3 n→∞o n
In Exercises use the formula for the th partial sum of a geometric seriesThe sphereflake shown below is a computer-generated fractal that was created by Eric Haines. The radius of the large sphere is 1. To the large sphere, nine spheres of radius 1/3 are attached. To each of these, nine spheres of
In Exercises consider making monthly deposits of P dollars in a savings account at an annual interest rate r. Use the results of Exercise 84 to find the balance A after t years when the interest is compounded (a) Monthly (b) ContinuouslyData from in Exercise 84When an employee receives a
In Exercises use the formula for the th partial sum of a geometric seriesThe winner of a $2,000,000 sweepstakes will be paid $100,000 per year for 20 years. The money earns 6% interest per year. The present value of the winnings isCompute the present value and interpret its meaning. n α(1 Σ'ar =
Consider the sequence(a) Compute the first five terms of this sequence.(b) Write a recursion formula for an for n ≥ 2.(c) Find √2, √√√2 + √2, 2 + 2 + √2,....
In Exercises use the formula for the nth partial sum of a geometric seriesYou go to work at a company that pays $0.01 for the first day, $0.02 for the second day, $0.04 for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working (a) 29 days(b) 30
Show that the converse of Theorem 9.1 is not true. THEOREM 9.1 Limit of a Sequence Let L be a real number. Let f be a function of a real variable such that lim f(x) = L. x-00 If {a} is a sequence such that f(n) = a, for every positive integer n, then lim a₁ = L. an 880
In Exercises use the formula for the nth partial sum of a geometric seriesWhen an employee receives a paycheck at the end of each month, P dollars is invested in a retirement account. These deposits are made each month for t years and the account earns interest at the annual percentage rate r. When
In Exercises determine the convergence or divergence of the series. 1 n=2 n(In n)³
A right triangle XYZ is shown above where |XY| = z and ∠X = θ̇. Line segments are continually drawn to be perpendicular to the triangle, as shown in the figure. (a) Find the total length of the perpendicular line segments |Yy₁| + x₁y₁| + x₁y₂ + in terms of z and θ̇.(b) Find the
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If {an} converges, then (a₂ − an+1) = 0. lim 00 n
In Exercises determine the convergence or divergence of the series. 18 n=2 In n
The sides of a square are 16 inches in length. A new square is formed by connecting the midpoints of the sides of the original square, and two of the triangles outside the second square are shaded (see figure). Determine the area of the shaded regions (a) When this process is continued five
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If {an} diverges and {bn} diverges, then {an, + bn} diverge.
In Exercises determine the convergence or divergence of the series. n=1 +)" 1 +
A fair coin is tossed repeatedly. The probability that the first head occurs on the nth toss is given by(a) Show that(b) The expected number of tosses required until the first head occurs in the experiment is given byIs this series geometric?(c) Use a computer algebra system to find the sum in part
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If {an} cTrueonverges, then {an,/n} converges to 0.
In Exercises the random variable n represents the number of units of a product sold per day in a store. The probability distribution of n is given by P(n). Find the probability that two units are sold in a given day [P(2)] and show that P(0) + P(1) + P(2) + P(3) + · · · = 1. P(n) || 1/2 3\3
In Exercises determine the convergence or divergence of the series. ÿ n=1 1 n² 1 3 n³
In an experiment, three people toss a fair coin one at a time until one of them tosses a head. Determine, for each person, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is 1.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If {an} converges to 3 and {bn} converges to 2, then (an + bn}converges to 5.
In Exercises determine the convergence or divergence of the series. n=1 n √n² + 1
In Exercises the random variable n represents the number of units of a product sold per day in a store. The probability distribution of n is given by P(n). Find the probability that two units are sold in a given day [P(2)] and show that P(0) + P(1) + P(2) + P(3) + · · · = 1. P(n) = 1/()* 22.
Compute the first six terms of the sequenceIf the sequence converges, find its limit. {an} 1 + n
In Exercises determine the convergence or divergence of the series. Σ (1.042)" n=0
Prove that if {sn} converges to L and L > 0, thenthere exists a number N such that sn, > 0 for n > N.
The ball in Exercise 75 takes the following times for each fall.Beginning with s2 the ball takes the same amount of time to bounce up as it does to fall, and so the total time elapsed before it comes to rest is given byFind this total time.Data from in Exercise 75A ball is dropped from a height of
In Exercises determine the convergence or divergence of the series. n=0 2/3
In Exercises determine the convergence or divergence of the series. ܝ3 n=1 1 10.95
Compute the first six terms of the sequence {an} = {n√n}. If the sequence converges, find its limit.
Repeat Exercise 73 when the percent of the revenue that is spent again in the city decreases to 60%.Data from in Exercise 73The total annual spending by tourists in a resort city is $200 million. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately
In Exercises determine the convergence or divergence of the series. 1 L 4 Sunn n=1 ก
A ball is dropped from a height of 16 feet. Each time it drops h feet, it rebounds 0.81h feet. Find the total distance traveled by the ball.
The total annual spending by tourists in a resort city is $200 million. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately 75% is again spent in the same city, and so on. Write the geometric series that gives the total amount of spending generated
The graphs of two sequences are shown in the figures. Which graph represents the sequence with alternating signs? Explain. 2 1 -1 -2 an 2 ●N n 2 1 - 1 an -2- 2 4 9 n
When the rate of inflation is 41/2% per year and the average price of a car is currently $25,000, the average price after n years is Pn = $25,000(1.045)n. Compute the averageprices for the next 5 years.
A government program that currently costs taxpayers $4.5 billion per year is cut back by 20 percent per year.(a) Write an expression for the amount budgeted for this program after n years.(b) Compute the budgets for the first 4 years.(c) Determine the convergence or divergence of the sequence of
A company buys a machine for $475,000 that depreciates at a rate of 30% per year. Find a formula for the value of the machine after n years. What is its value after 5 years?
Give an example of a sequence satisfying the condition or explain why no such sequence exists.(a) A monotonically increasing sequence that converges to 10(b) A monotonically increasing bounded sequence that does not converge(c) A sequence that converges to 3/4(d) An unbounded sequence that
In Exercises determine the convergence or divergence of the series. n=1 1 3n - 2
An electronic games manufacturer producinga new product estimates the annual sales to be 8000 units.Each year, 5% of the units that have been sold will becomeinoperative. So, 8000 units will be in use after 1 year,[8000+ 0.95(8000)] units will be in use after 2 years, and soon. How many units will
Consider the series(a) Determine the convergence or divergence of the series for x = 1.(b) Determine the convergence or divergence of the series for x = 1/e.(c) Find the positive values of for which the series converges. Ž xln n n=2
In Exercises (a) Find the common ratio of the geometric series(b) Write the function that gives the sum of the series(c) Use a graphing utility to graph the function and the partial sums S3 and S5 What do you notice? x² 1- + 2 4 8 +
Is it possible for a sequence to converge to two different numbers? If so, give an example. If not, explain why not.
Is the infinite seriesconvergent? Prove your statement. 8 n=1 1 n(n+1)/n
In Exercises find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. n=0 (-1)" xn
Prove that the seriesconverges. n= 1 1+2+3+.. + n
Let {an} be a monotonic sequence such that an, ≤ 1. Discuss the convergence of {an}.When {a} converges, what can you conclude about its limit?
In Exercises (a) Use Theorem 9.5 to show that the sequence with the given th term converges(b) Use a graphing utility to graph the first 10 terms of the sequence and find its limit. Data from in Theorem 9.5 THEOREM 9.5 Bounded Monotonic Sequences If a sequence {a} is bounded and
Let {an} be an increasingsequence such that 2 ≤ an, ≤ 4. Explain why {an} has a limit.What can you conclude about the limit?
In Exercises find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. n=1 (x - 1)n
In Exercises find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. n=0 2 n X
In Exercises (a) Use Theorem 9.5 to show that the sequence with the given th term converges(b) Use a graphing utility to graph the first 10 terms of the sequence and find its limit.Data from in Theorem 9.5 THEOREM 9.5 Bounded Monotonic Sequences If a sequence {a} is bounded and monotonic, then
In Exercises use the result of Exercise 53 to find N such that RN ≤ 0.001 for the convergent series.Data from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such that an = ƒ(n). Prove that if the seriesconverges to S, then the remainder RN S - SN
In Exercises use the result of Exercise 53 to find N such that RN ≤ 0.001 for the convergent series.Data from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such that an = ƒ(n). Prove that if the seriesconverges to S, then the remainder RN S - SN
In Exercises find all values of x for which the series converges. For these values of x, write the sum of the series as a function of x. ∞ n=1 (3x)n
In Exercises (a) Use Theorem 9.5 to show that the sequence with the given th term converges(b) Use a graphing utility to graph the first 10 terms of the sequence and find its limit.Data from in Theorem 9.5 THEOREM 9.5 Bounded Monotonic Sequences If a sequence {a} is bounded and monotonic, then
Suppose thatΣ an is a series with positive terms. Prove that if Σ an converges, then Σ sin an also converges.
In Exercises use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation.Data from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such
Explain any differences among the following series.(a)(b)(c) 00 Σαπ a n=1
In Exercises use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation.Data from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such
In Exercises determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. = sin an nπ 6
Describe the difference between lim an n→∞ 5 and a₁ = 5. an n=1
Suppose that Σ an and Σ bn are series with positive an n→∞ b n terms. Prove that if and Σ bn converges, Σ an also converges. an lim n→∞0 b₂ n. = 0
In Exercises use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation.Data from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such
In Exercises determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. an 2/3
In Exercises use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation.Data from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such
Find two series that demonstrate the result of Exercise 55.Data from in Exercise 55Prove that if the nonnegative seriesconverge, then so does the series n=1 and n=1 bn
In Exercises use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation.Data from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such
In Exercises determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. an || n
Find two series that demonstrate the result of Exercise 56.Data from in Exercise 56prove that if the nonnegative seriesconverges, then so does the series 18 n=1 an
In Exercises determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. an = 2 3 n
In Exercises determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. a n ne-n/2
Use the result of Exercise 55 to prove that if the nonnegative seriesconverges, then so does the seriesData from in Exercise 55Prove that if the nonnegative seriesconverge, then so does the series 18 n=1 an
State the nth-Term Test for Divergence.
Define a geometric series, state when it converges, and give the formula for the sum of a convergent geometric series.
In Exercises determine the convergence or divergence of the series. n=1 In (n+1) n
In Exercises determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. || 3n n + 2
Prove that if the nonnegative seriesconverge, then so does the series n=1 and n=1 bn
In Exercises use the result of Exercise 53 to approximate the sum of the convergent series using the indicated number of terms. Include an estimate of the maximum error for your approximation.Data from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such
In Exercises determine the convergence or divergence of the series. n=1 arctan n
In Exercises determine whether the sequence with the given nth term is monotonic and whether it is bounded. Use a graphing utility to confirm your results. an = 4. n
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If 0 < a ≤ b, and n=1 b, diverges, then n Σ an diverges. n=1
Show that the result of Exercise 53 can be written asData from in Exercise 53 Let ƒ be a positive, continuous, and decreasing function for x ≥ 1, such that an = ƒ(n). Prove that if the seriesconverges to S, then the remainder RN S - SN is bounded by N h=1 | = W < n=1 N an = Σ an
In Exercises determine the convergence or divergence of the series. Σ ρτη n=1
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If 0 < a ≤ b, and n n=1 a, diverges, then n= b diverges.
Let ƒ be a positive, continuous, and decreasingfunction for x ≥ 1, such that an = ƒ(n). Prove that if the seriesconverges to S, then the remainder RN S - SN is bounded by Σ n=1 απ

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