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mathematics
calculus early transcendentals
Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. k2 + k – 1 k=1 k* + 4k² – 3 8.
Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. k=1 k² + 4
Use the Root Test to determine whether the following series converge. Σ k k=1 e 8.
Use the Root Test to determine whether the following series converge. (4) 3 (4) 2 4 2 3
Use the Root Test to determine whether the following series converge. k 1 In (k + 1). k=1 M:
Use the Root Test to determine whether the following series converge. 2k2 k k + 1 k=1
Use the Root Test to determine whether the following series converge. 3 Σ k k=1
Use the Root Test to determine whether the following series converge. Σ ok k=1 8.
Use the Root Test to determine whether the following series converge. k 2k Σ k + 1 k=1
Use the Root Test to determine whether the following series converge. 00 10k + k Σ 9k3 + k + 1 ,3 |k=1
Use the Ratio Test to determine whether the following series converge. 4 |2 + 16 8. 16 81 256
Use the Ratio Test to determine whether the following series converge. (k!)² Σ 00 (2k)! k=1
Use the Ratio Test to determine whether the following series converge. k6 k! k=1
Use the Ratio Test to determine whether the following series converge. 2k Σ k 99 k=1
Use the Ratio Test to determine whether the following series converge. kk k! k=1
Use the Ratio Test to determine whether the following series converge. -k Σke k=1
Use the Ratio Test to determine whether the following series converge. kk 2k k=1
Use the Ratio Test to determine whether the following series converge. 4k k=1
Use the Ratio Test to determine whether the following series converge. 2k k! k=1 8.
Use the Ratio Test to determine whether the following series converge. Σ k! k=1
Do the tests discussed in this section tell you the value of the series? Explain.
Explain why, with a series of positive terms, the sequence of partial sums is an increasing sequence.
What tests are best for the series ∑ak when ak is a rational function of k?
What test is advisable if a series of positive terms involves a factorial term?
What is the first test you should use in analyzing the convergence of a series?
Explain how the Limit Comparison Test works.
Explain how the Root Test works.
Explain how the Ratio Test works.
The Fibonacci sequence {1, 1, 2, 3, 5, 8, 13, . . .} is generated by the recurrence relationfn + 1 = fn + fn - 1, for n = 1, 2, 3, . . , where f0 = 1, f1 = 1.a. It can be shown that the sequence of ratios of successive terms of the sequence has a limit φ. Divide both sides of the recurrence
Consider a wedding cake of infinite height, each layer of which is a right circular cylinder of height 1. The bottom layer of the cake has a radius of 1, the second layer has a radius of 1/2, the third layer has a radius of 1/3, and the nth layer has a radius of 1/n (see figure). a. To
Consider a set of identical dominoes that are 2 inches long. The dominoes are stacked on top of each other with their long edges aligned so that each domino overhangs the one beneath it as far as possible (see figure).a. If there are n dominoes in the stack, what is the greatest distance that the
a. Sketch the function f(x) = 1/x on the interval [1, n + 1], where n is a positive integer. Use this graph to verify thatb. Let Sn be the sum of the first n terms of the harmonic series, so part (a) says ln (n + 1) < Sn < 1 + ln n. Define the new sequence {En} by En = Sn - ln (n + 1),
Consider the sequence {xn} defined for n = 1, 2, 3, . . . bya. Write out the terms x1, x2, x3.b. Show that 1/2 ≤ xn < 1, for n = 1, 2, 3, . . .c. Show that xn is the right Riemann sum for using n subintervals.d. Conclude that 2n 1 2n n + 1 n + 2 k k=n+1 || xp,
Consider the sequence {Fn} defined byfor n = 0, 1, 2, c. When n = 0, the series is a p-series, and we have F0 = π2/6 (Exercises 65 and 66).a. Explain why {Fn} is a decreasing sequence.b. Plot {Fn}, for n = 1, 2, . . . , 20.c. Based on your experiments, make a conjecture aboutData from Exercise
In 1734, Leonhard Euler informally proved that An elegant proof is outlined here that uses the inequalitycot2 x < 1/x2 < 1 + cot2 x (provided that 0 < x < π/2) and the identitya. Show that b. Use the inequality in part (a) to show thatc. Use the Squeeze Theorem to conclude that .2 TI 1
The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined byWhen x is a real number, the zeta function becomes a p-series. For even positive integers p, the value ofis known exactly. For example,Use the estimation
The prime numbers are those positive integers that are divisible by only 1 and themselves (for example, 2, 3, 5, 7, 11, 13, . . ). A celebrated theorem states that the sequence of prime numbers {pk} satisfies Show that diverges, which implies that the series diverges. lim pr/(k
series Prove that if ∑ak diverges, then ∑cak also diverges, where c º 0 is a constant.
Use the ideas in the proof of Property 1 of Theorem 8.13 to prove Property 2 of Theorem 8.13.
Give an argument similar to that given in the text for the harmonic series to show thatdiverges. k=1 Vk
Find a series thata. Converges faster than ∑ 1/k2 but slower than ∑ 1/k3.b. Diverges faster than ∑ 1/k but slower than ∑ 1/√k.c. Converges faster than ∑ 1/k ln2 k but slower than ∑ 1/k2.
Consider the series
Consider the series where p is a real number.a. Use the Integral Test to determine the values of p for which this series converges.b. Does this series converge faster for p = 2 or p = 3? Explain. k=2
Determine whether the following series converge or diverge. Σ ? k=2 k (In k)²
Determine whether the following series converge or diverge. 2k + 3k 4k k=1
Determine whether the following series converge or diverge. Vk? + 1 k=1
Determine whether the following series converge or diverge. 10 Σ k=0 k + 9 .2
Determine whether the following series converge or diverge. Σ (3k + 1)(3k + 4) k=1
Determine whether the following series converge or diverge. k + 1 k k=1
Determine whether the following statements are true and give an explanation or counterexample.a. Ifconverges, thenconverges.b. If diverges, then diverges.c. If ∑ak converges, then ∑(ak + 0.0001) also converges.d. If ∑pk diverges, then ∑(p + 0.001)k diverges, for a fixed real number p.e. If
Use the properties of infinite series to evaluate the following series. 2 – 3k 6k k=0 8.
Use the properties of infinite series to evaluate the following series.
Use the properties of infinite series to evaluate the following series. 3 (0.2)* + (0.8 k=0
Use the properties of infinite series to evaluate the following series. k 5 3 Σ 5 (9 3 6. . |k=1 8.
Use the properties of infinite series to evaluate the following series. 3 + 3 k=1 8.
Use the properties of infinite series to evaluate the following series. ¿O) - O) k k 3 7. k=0
Use the properties of infinite series to evaluate the following series. -k Зе t k=2
Use the properties of infinite series to evaluate the following series. 4 Σ 12k k=1 8.
Consider the following convergent series.a. Find an upper bound for the remainder in terms of n.b. Find how many terms are needed to ensure that the remainder is less than 10-3.c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.d. Find an interval in which the
Consider the following convergent series.a. Find an upper bound for the remainder in terms of n.b. Find how many terms are needed to ensure that the remainder is less than 10-3.c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.d. Find an interval in which the
Consider the following convergent series.a. Find an upper bound for the remainder in terms of n.b. Find how many terms are needed to ensure that the remainder is less than 10-3.c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.d. Find an interval in which the
Consider the following convergent series.a. Find an upper bound for the remainder in terms of n.b. Find how many terms are needed to ensure that the remainder is less than 10-3.c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.d. Find an interval in which the
Consider the following convergent series.a. Find an upper bound for the remainder in terms of n.b. Find how many terms are needed to ensure that the remainder is less than 10-3.c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.d. Find an interval in which the
Consider the following convergent series.a. Find an upper bound for the remainder in terms of n.b. Find how many terms are needed to ensure that the remainder is less than 10-3.c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.d. Find an interval in which the
Consider the following convergent series.a. Find an upper bound for the remainder in terms of n.b. Find how many terms are needed to ensure that the remainder is less than 10-3.c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.d. Find an interval in which the
Consider the following convergent series.a. Find an upper bound for the remainder in terms of n.b. Find how many terms are needed to ensure that the remainder is less than 10-3.c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.d. Find an interval in which the
Determine the convergence or divergence of the following series. 3, k=1 27k2
Determine the convergence or divergence of the following series. Vk k=1
Determine the convergence or divergence of the following series. 2k k=1 8.
Determine the convergence or divergence of the following series. ο0 Σ (k – 2)4 k=3
Determine the convergence or divergence of the following series. ke kT k=2
Determine the convergence or divergence of the following series. 1 k=1 k10
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. k (k2 + 1) 3 k=1 8.
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. |sin k | k? k=1 8.
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. Σ k=3 k (In k) In In k 8.
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. k k=1 et
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. Σ A k(In k)? k=2
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. 00 Σ k=0 Vk + 8
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. Σ Vk + 10 k=1
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. –2k² Z ke k=1
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. Σ k=1 Vk? + 4 -2
Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply.
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. .3 k! k=1
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 00 1/k k=1
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. Vk? + 1 k k=1
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. Vk k=2 In1º k
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. k3 Σ k=1 k + 1
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. Σ 1000 + k k=0
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 12 2* k=1
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 00 k Σ k=2 In k
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. k Σ k=1 k2 + 1
Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. k Σ 2k + 1 k=0
If a series of positive terms converges, does it follow that the remainder Rn must decrease to zero as n →∞? Explain.
Define the remainder of an infinite series.
Explain why the sequence of partial sums for a series with positive terms is an increasing sequence.
For what values of p does the series converge (initial index is 10)? For what values of p does it diverge? Σ kP k=10
For what values of p does the seriesconverge? For what values of p does it diverge? kP k=1
Find the nth-order Taylor polynomial for the following functions centered at the given point a.f(x) = sin 2x, n = 3, a = 0
Determine whether the following statements are true and give an explanation or counterexample.a. Let pn be the nth-order Taylor polynomial for f centered at 2. The approximation p3(2.1) ≈ f(2.1) is likely to be more accurate than the approximation p2(2.2) ≈ f(2.2).b. If the Taylor series for f
Newton discovered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point.a. Referring to the figure, show that x = sin s or s = sin-1 x.b. The area of a circular sector of radius r subtended by an angle θ is 1/2 r2 θ. Show that the area of the
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