Explanation Given information: The infinite series is ∑ k = 2 ∞ 1 k ( ln k ) 2 . Explanation: Here, f ( x ) = 1 x ( ln x ) 2 is a positive and continuous for x ≥ 2 . f ' ( x ) = d d x ( x ( ln x ) 2 ) − 1 Substitute u = x ( ln x ) 2 then use general power rule of derivative, d d x ( x ( ln x ) 2 ) − 1 = d d u ( u − 1 ) ⋅ d d x ( x ( ln x ) 2 ) = − 1 u 2 ⋅ ( ( ln x ) 2 + 2 ln x ) Substituting u = x ( ln x ) 2 again, it gives = − 1 ( x ( ln x ) 2 ) 2 ⋅ ( ( ln x ) 2 + 2 ln x ) Simplifying, ⇒ f ' ( x ) = − ln x + 2 x 2 ( ln x ) 3 So, f ' ( x ) is negative and hence f ( x ) = 1 x ( ln x ) 2 is a decreasing function. Integral test, Let f ( x ) be continuous, decreasing, and positive for x ≥ 1 . Then, the infinite series ∑ k = 1 ∞ f ( k ) is convergent if the improper integral ∫ 1 ∞ f ( x ) d x is convergent, and the infinite series is divergent if the improper integral is divergent

12th Edition

ISBN: 9780137590810

Author: Larry J. Goldstein, David C. Lay, David I. Schneider, Nakhle H. Asmar, William Edward Tavernetti

Publisher: PEARSON

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Chapter 11.4, Problem 9E

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Theinfinite series ∑k=2∞1k(ln k)2 is convergent or divergent by using the integral test.

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Chapter 11.4, Problem 8E

Chapter 11.4, Problem 10E

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Chapter 11 Solutions

Calculus & Its Applications

Ch. 11.1 - Determine the third Taylor polynomial of f(x)=cosx...Ch. 11.1 - Prob. 2CYU Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E

Ch. 11.1 - Prob. 9E Ch. 11.1 - Prob. 10E Ch. 11.1 - Prob. 11E Ch. 11.1 - Prob. 12E Ch. 11.1 - Prob. 13E Ch. 11.1 - Prob. 14E Ch. 11.1 - Prob. 15E Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 17E Ch. 11.1 - Prob. 18E Ch. 11.1 - Determine the third and fourthTaylor polynomial...Ch. 11.1 - Prob. 20E Ch. 11.1 - Prob. 21E Ch. 11.1 - Prob. 22E Ch. 11.1 - Prob. 23E Ch. 11.1 - Prob. 24E Ch. 11.1 - Prob. 25E Ch. 11.1 - Prob. 26E Ch. 11.1 - Prob. 27E Ch. 11.1 - Prob. 28E Ch. 11.1 - Prob. 29E Ch. 11.1 - Prob. 30E Ch. 11.1 - Graph the function Y1=11x and its fourth Taylor...Ch. 11.1 - Prob. 32E Ch. 11.1 - Prob. 33E Ch. 11.1 - Prob. 34E Ch. 11.2 - Prob. 1CYU Ch. 11.2 - Prob. 2CYU Ch. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - Prob. 3E Ch. 11.2 - Prob. 4E Ch. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - Prob. 6E Ch. 11.2 - Prob. 7E Ch. 11.2 - Prob. 8E Ch. 11.2 - Sketch the graph of y=x3+2x+2, and use the...Ch. 11.2 - Prob. 10E Ch. 11.2 - Prob. 11E Ch. 11.2 - Prob. 12E Ch. 11.2 - Prob. 13E Ch. 11.2 - Internet Rate of Return An investor buys a bond...Ch. 11.2 - Prob. 15E Ch. 11.2 - Prob. 16E Ch. 11.2 - Prob. 17E Ch. 11.2 - Prob. 18E Ch. 11.2 - Prob. 19E Ch. 11.2 - Prob. 20E Ch. 11.2 - Prob. 21E Ch. 11.2 - Figure 9contains the graph of the function...Ch. 11.2 - Prob. 23E Ch. 11.2 - Prob. 24E Ch. 11.2 - Exercises 25 and 26 present two examples in which...Ch. 11.2 - Prob. 26E Ch. 11.2 - Prob. 27E Ch. 11.2 - Prob. 28E Ch. 11.2 - Prob. 29E Ch. 11.2 - Prob. 30E Ch. 11.3 - Determine the sum of the geometric series...Ch. 11.3 - Prob. 2CYU Ch. 11.3 - Determine the sums of the following geometric...Ch. 11.3 - Prob. 2E Ch. 11.3 - Prob. 3E Ch. 11.3 - Determine the sums of the following geometric...Ch. 11.3 - Prob. 5E Ch. 11.3 - Determine the sums of the following geometric...Ch. 11.3 - Prob. 7E Ch. 11.3 - Prob. 8E Ch. 11.3 - Prob. 9E Ch. 11.3 - Prob. 10E Ch. 11.3 - Prob. 11E Ch. 11.3 - Prob. 12E Ch. 11.3 - Prob. 13E Ch. 11.3 - Prob. 14E Ch. 11.3 - Prob. 15E Ch. 11.3 - Sum an appropriate infinite series to find the...Ch. 11.3 - Prob. 17E Ch. 11.3 - Sum an appropriate infinite series to find the...Ch. 11.3 - Prob. 19E Ch. 11.3 - Prob. 20E Ch. 11.3 - Prob. 21E Ch. 11.3 - Prob. 22E Ch. 11.3 - Prob. 23E Ch. 11.3 - The Multiplier Effect Compute the effect of a 20...Ch. 11.3 - Perpetuity Consider a perpetuity that promises to...Ch. 11.3 - Prob. 26E Ch. 11.3 - Bonus plus Taxes on Taxes A generous corporation...Ch. 11.3 - Total Distance Travelled by a Bouncing Ball The...Ch. 11.3 - Elimination of a Drug A patient receives 6 mg of a...Ch. 11.3 - Elimination of a Drug A patient receives 2 mg of a...Ch. 11.3 - Drug Dosage A patient receives M mg of a certain...Ch. 11.3 - Drug Dosage A patient receives M mg of a certain...Ch. 11.3 - Prob. 33E Ch. 11.3 - The infinite series a1+a2+a3+ has partial sums...Ch. 11.3 - Prob. 35E Ch. 11.3 - Prob. 36E Ch. 11.3 - Prob. 37E Ch. 11.3 - Determine the sums of the following infinite...Ch. 11.3 - Prob. 39E Ch. 11.3 - Prob. 40E Ch. 11.3 - Prob. 41E Ch. 11.3 - Prob. 42E Ch. 11.3 - Prob. 43E Ch. 11.3 - Prob. 44E Ch. 11.3 - Prob. 45E Ch. 11.3 - Prob. 46E Ch. 11.3 - Prob. 47E Ch. 11.3 - Prob. 48E Ch. 11.3 - Prob. 49E Ch. 11.3 - In Exercises 49 and 50, convince yourself that the...Ch. 11.4 - What is the improper integral associated with the...Ch. 11.4 - Prob. 2CYU Ch. 11.4 - Prob. 1E Ch. 11.4 - Prob. 2E Ch. 11.4 - Prob. 3E Ch. 11.4 - Prob. 4E Ch. 11.4 - Prob. 5E Ch. 11.4 - Prob. 6E Ch. 11.4 - Prob. 7E Ch. 11.4 - Prob. 8E Ch. 11.4 - Prob. 9E Ch. 11.4 - Prob. 10E Ch. 11.4 - In Exercises 116, use the integral test to...Ch. 11.4 - Prob. 12E Ch. 11.4 - Prob. 13E Ch. 11.4 - Prob. 14E Ch. 11.4 - Prob. 15E Ch. 11.4 - In Exercises 116, use the integral test to...Ch. 11.4 - Prob. 17E Ch. 11.4 - Prob. 18E Ch. 11.4 - Prob. 19E Ch. 11.4 - Prob. 20E Ch. 11.4 - In Excercises 2126, use the comparison test to...Ch. 11.4 - Prob. 22E Ch. 11.4 - Prob. 23E Ch. 11.4 - Prob. 24E Ch. 11.4 - Prob. 25E Ch. 11.4 - Prob. 26E Ch. 11.4 - Prob. 27E Ch. 11.4 - Prob. 28E Ch. 11.4 - Prob. 29E Ch. 11.4 - Prob. 30E Ch. 11.4 - Use Exercise 29 to show that the series...Ch. 11.4 - Use Exercise 30 to show that the series k=13k2 is...Ch. 11.5 - Find the Taylor series expansion of sinx at x=0.Ch. 11.5 - Find the Taylor series expansion of cosx at x=0.Ch. 11.5 - Prob. 3CYU Ch. 11.5 - Prob. 4CYU Ch. 11.5 - Prob. 1E Ch. 11.5 - Prob. 2E Ch. 11.5 - Prob. 3E Ch. 11.5 - In Exercises 14, find the Taylor series at x=0 of...Ch. 11.5 - Prob. 5E Ch. 11.5 - Prob. 6E Ch. 11.5 - Prob. 7E Ch. 11.5 - Prob. 8E Ch. 11.5 - Prob. 9E Ch. 11.5 - In Exercises 520, find the Taylor series at x=0 of...Ch. 11.5 - Prob. 11E Ch. 11.5 - Prob. 12E Ch. 11.5 - Prob. 13E Ch. 11.5 - Prob. 14E Ch. 11.5 - In Exercises 520, find the Taylor series at x=0 of...Ch. 11.5 - Prob. 16E Ch. 11.5 - Prob. 17E Ch. 11.5 - Prob. 18E Ch. 11.5 - Prob. 19E Ch. 11.5 - In Exercises 520, find the Taylor series at x=0 of...Ch. 11.5 - Find the Taylor series of xex2 at x=0.Ch. 11.5 - Prob. 22E Ch. 11.5 - Prob. 23E Ch. 11.5 - Prob. 24E Ch. 11.5 - Prob. 25E Ch. 11.5 - Prob. 26E Ch. 11.5 - Prob. 27E Ch. 11.5 - Prob. 28E Ch. 11.5 - Prob. 29E Ch. 11.5 - Prob. 30E Ch. 11.5 - Prob. 31E Ch. 11.5 - Prob. 32E Ch. 11.5 - Prob. 33E Ch. 11.5 - The Taylor series at x=0 for 1+x21x is...Ch. 11.5 - Prob. 35E Ch. 11.5 - Prob. 36E Ch. 11.5 - Prob. 37E Ch. 11.5 - Prob. 38E Ch. 11.5 - In Exercises 3840, find the infinite series that...Ch. 11.5 - Prob. 40E Ch. 11.5 - Prob. 41E Ch. 11.5 - Prob. 42E Ch. 11.5 - Prob. 43E Ch. 11.5 - Prob. 44E Ch. 11.5 - Prob. 45E Ch. 11.5 - Prob. 46E Ch. 11 - Prob. 1CYU Ch. 11 - Prob. 2CYU Ch. 11 - Prob. 3CYU Ch. 11 - Prob. 4CYU Ch. 11 - Prob. 5CYU Ch. 11 - Prob. 6CYU Ch. 11 - What is meant by the sum of a convergent infinite...Ch. 11 - Prob. 8CYU Ch. 11 - Prob. 9CYU Ch. 11 - Prob. 10CYU Ch. 11 - Prob. 11CYU Ch. 11 - Prob. 1RE Ch. 11 - Prob. 2RE Ch. 11 - Prob. 3RE Ch. 11 - Prob. 4RE Ch. 11 - Prob. 5RE Ch. 11 - Prob. 6RE Ch. 11 - Prob. 7RE Ch. 11 - Prob. 8RE Ch. 11 - 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Theinfinite series ∑ k = 2 ∞ 1 k ( ln k ) 2 is convergent or divergent by using the integral test.

Solution Summary: The author explains that the infinite series displaystyle 'underset'k=2'oversetinfty'sum is convergent or divergent.

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Theinfinite series ∑k=2∞1k(ln k)2 is convergent or divergent by using the integral test.

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Chapter 11 Solutions