a. Explanation Result used: Comparison test: Consider the series ∑ k = 1 ∞ a k , where a k > 0 . If a k ≤ b k and ∑ k = 1 ∞ b k converges, then the series converges and if b k ≤ a k and ∑ k = 1 ∞ b k diverges, then the series diverges. Limit Comparison test: Consider the series ∑ k = 1 ∞ a k , where a k > 0 , b k > 0 . If 0 ≤ lim k → ∞ a k b k < ∞ and ∑ k = 1 ∞ b k converges, then the series converges and if lim k → ∞ a k b k > 0 and ∑ k = 1 ∞ b k diverges, then the series diverges. p-series: The series ∑ k = 1 ∞ 1 k p converges for p > 1 and diverges for p ≤ 1 . Calculation: The given statement is false, which can proved by the example shown below. Consider the series ∑ k = 1 ∞ 4 k 2 − k − 1 3 k 4 − 2 k + 1 . Simplify the series ∑ k = 1 ∞ 4 k 2 − k − 1 3 k 4 − 2 k + 1 as follows. ∑ k = 1 ∞ 4 k 2 − k − 1 3 k 4 − 2 k + 1 = ∑ k = 1 ∞ a k , where a k = 4 k 2 − k − 1 3 k 4 − 2 k + 1 . Suppose b k = k 2 k 4 . Clearly, by the p -series the series ∑ k = 1 ∞ 1 k 2 converges and so the series ∑ k = 1 ∞ b k converges. Compute the value of lim k → ∞ a k b k b. To determine Whether the statement “The series ∑ k = 3 ∞ 1 k ln p k converges for the same values of p as the series ∑ k = 3 ∞ 1 k p ” is true or false. c. To determine Whether the statement “Both the Ratio Test and the Root Test can be applied conclusively to a geometric series” is true or false. d. To determine Whether the statement “The Alternating Series Test can be used to show that some series diverge” is true or false.

Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package

3rd Edition

ISBN: 9780136207764

Author: Briggs, William, Cochran, Lyle, Gillett, Bernard, SCHULZ, Eric

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 10.8, Problem 87E

To determine

Whether the statement “If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series” is true or false.

To determine

Whether the statement “The series ∑k=3∞1klnpk converges for the same values of p as the series ∑k=3∞1kp” is true or false.

To determine

Whether the statement “Both the Ratio Test and the Root Test can be applied conclusively to a geometric series” is true or false.

To determine

Whether the statement “The Alternating Series Test can be used to show that some series diverge” is true or false.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.8, Problem 86E

Chapter 10.8, Problem 88E

Students have asked these similar questions

Evaluate the double integral ' √ √ (−2xy² + 3ry) dA R where R = {(x,y)| 1 ≤ x ≤ 3, 2 ≤ y ≤ 4} Double Integral Plot of integrand and Region R N 120 100 80- 60- 40 20 -20 -40 2 T 3 4 5123456 This plot is an example of the function over region R. The region and function identified in your problem will be slightly different. Answer = Round your answer to four decimal places.

Find Te²+ dydz 0 Write your answer in exact form.

xy² Find -dA, R = [0,3] × [−4,4] x²+1 Round your answer to four decimal places.

Chapter 10 Solutions

Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package

Ch. 10.1 - Find a10 for the sequence {1, 4, 7, 10, ...} using...Ch. 10.1 - Find an explicit formula for the sequence {1, 3,...Ch. 10.1 - Reasoning as in Example 7, what is the value of...Ch. 10.1 - Do the series k=11 and k=1k converge or diverge?Ch. 10.1 - Find the first four terms of the sequence of...Ch. 10.1 - Define sequence and give an example.Ch. 10.1 - Suppose the sequence {an} is defined by the...Ch. 10.1 - Suppose the sequence {an} is defined by the...Ch. 10.1 - Find two different explicit formulas for the...Ch. 10.1 - Find two different explicit formulas for the...

Additional Math Textbook Solutions

Find more solutions based on key concepts

To explain whether the statement is always, sometimes or never true.

Pre-Algebra Student Edition

Derivatives of inverse sine Evaluate the derivatives of the following functions. 7. f(x) = sin1 2x

Calculus: Early Transcendentals (2nd Edition)

Assessment 1-1A How many triangles are in the following figure?

A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)

Fill in each blank so that the resulting statement is true. The quadratic function f(x)=a(xh)2+k,a0, is in ____...

Algebra and Trigonometry (6th Edition)

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean. 25. (12.0, 14....

Elementary Statistics: Picturing the World (7th Edition)

Explain the meaning of the term “statistically significant difference” in statistics terminology.

Intro Stats, Books a la Carte Edition (5th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.