Explanation Given: The series is ∑ k = 4 ∞ a k = ∑ k = 4 ∞ 2 k 2 − 10 . Result used: “Suppose f is a continuous, positive, decreasing function, for x ≥ 1 , and let a k = f ( k ) for k = 1 , 2 , 3 , ⋯ , then ∑ k = 1 ∞ a k and ∫ 1 ∞ f ( x ) d x either both converge or both diverge. Calculation: Consider the function from given series 2 k 2 − 10 . The derivative of the function f ( x ) = 2 ( x 2 − 10 ) − 1 is obtained as follows, f ′ ( x ) = − 2 ( x 2 − 10 ) − 1 − 1 = − 2 ( x 2 − 10 ) − 2 Clearly, the function f ′ ( x ) is continuous, positive and decreasing for x ≥ 5 . Use the Result (1), the series is divergent if the improper integral ∫ 4 ∞ 2 x 2 − 10 d x is divergent. By the definition, the improper integral is divergent if the limit does not exist. Compute ∫ 4 ∞ 2 x 2 − 10 d x as shown below. ∫ 4 ∞ 2 x 2 − 10 d x = lim t → ∞ [ ∫ 4 t 2 x 2 − 10 d x ] (1) Obtain ∫ 4 t 2 x 2 − 10 d x . Consider the definite integral ∫ 2 x 2 − 10

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 8, Problem 34RE

To determine

Whether the series converges or diverges.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8, Problem 33RE

Chapter 8, Problem 35RE

Students have asked these similar questions

20 (-3)* B. Find the interval of convergence of the power series, sum of the series as a function. and find the

Find the interval of convergence for the given power series. (x - 4)" Σ n(- 9)" n=1 The series is convergent from x = left end included (enter Y or N): to x = right end included (enter Y or N): M C ㅈ # $ A de L % 5 6 D 8 7 8 9 #

n3 + 2 1. (15 p) Determine if the series converges or diverges. +3 n=1 Give reason for your answer.

Chapter 8 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 8.1 - Define sequence and give an example.Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Suppose the sequence {an} is defined by the...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Given the series k=1k, evaluate the first four...Ch. 8.1 - The terms of a sequence of partial sums are...Ch. 8.1 - Consider the infinite series k=11k. Evaluate the...Ch. 8.1 - Explicit formulas Write the first four terms of...Ch. 8.1 - Explicit formulas Write the first four terms of...

Additional Math Textbook Solutions

Find more solutions based on key concepts

1. On a real number line the origin is assigned the number _____ .

Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)

The inequality 1x3 can be written in interval notation as _______________. (pp. A76-A77)

Precalculus

The given statement is true or false: “A function is a relation between two sets D and R so that each element x...

Precalculus

Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify t...

University Calculus: Early Transcendentals (4th Edition)

Find the slopes of the following lines. The line going through the points (2,5)and(2,8).

Calculus & Its Applications (14th Edition)

4 3 = ; 8 2/3 = ; 3 2 = . (pp. A8-A9 and pp, A89-A91)

Precalculus (10th 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.

Whether the series converges or diverges.

Solution Summary: The author analyzes whether the series converges or diverges. The derivative of the function f(x) is continuous, positive and decreasing.

Videos

Whether the series converges or diverges.

Want to see the full answer?

Chapter 8 Solutions

Additional Math Textbook Solutions