Explanation Given: The series is ∑ k = 0 ∞ k ! p k ( k + 1 ) k . Result used: Let the series ∑ a k of non-negative terms and r = lim k → ∞ | a k + 1 a k | . (i) If 0 ≤ r < 1 , the series converges. (ii) If r > 1 ( including r = ∞ ) , the series diverges. (iii) If r = 1 , the test is inconclusive Theorem used: The geometric series ∑ k = 0 ∞ a r k converges if | r | < 1 . Calculation: From the given series, the k th term is, a k = k ! p k ( k + 1 ) k . Thus, the ( k + 1 ) th term is, a k + 1 = ( k + 1 ) ! p k + 1 ( ( k + 1 ) + 1 ) k + 1 . Obtain the limit of | a k + 1 a k | . lim k → ∞ | a k + 1 a k | = lim k → ∞ | ( k + 1 ) ! p k + 1 ( ( k + 1 ) + 1 ) k + 1 k ! p k ( k + 1 ) k | = lim k → ∞ | ( k + 1 ) ! p k + 1 ( k + 1 ) k k ! p k ( k + 2 ) k + 1 | = lim k → ∞ k ! p ( k + 1 ) k + 1 k ! ( k + 2 ) k + 1 = lim k → ∞ p ( k + 1 k + 1 ) k + 1 That is, lim k → ∞ | a k + 1 a k | = lim k → ∞ p 1 ( 1 + 1 k + 1 ) k + 1 (1) Obtain the limit of ( 1 + 1 k + 1 ) k + 1

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 8.5, Problem 74E

To determine

To find: The values of the parameter p>0 for the series converges.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.5, Problem 73E

Chapter 8.5, Problem 75E

Students have asked these similar questions

Binomial seriesa. Find the first four nonzero terms of the binomial series centered at 0 for the given function.b. Use the first four terms of the series to approximate the given quantity.

Σ 1x Represent the function 2 + x series f(x) = > as a power n=0 C = C2 C3 = C4 Find the radius of convergence R=

(Review) Determine whether the series E(-1)n+1 n3+5 Converges absolutely, converges conditionally, or diverges. n=1 a. converges absolutely b. converges conditionally c. diverges d. cannot be determined Let f(x) = 1 Which of the following are true? (2 marks) 1+2x² a) f(x) = E(-1)" 2"x²n if – 1< x < 1. n=0 b) f(x) = E(-1)" 2"x²n if – < x < %: n=0 c) T3,0(x) = 1 – 2x2 . d) f"(0) = -2³ · 3!. e) ſi H da 1+2x² E (-1)" 2"[2²n – 1). n=0

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

76. Dew Point and Altitude The dew point decreases as altitude increases. If the dew point on the ground is 80°...

College Algebra with Modeling & Visualization (5th Edition)

In Exercises 13–16, find the margin of error for the values of c, ?, and n. 13. c = 0.95, ? = 5.2, n = 30

Elementary Statistics: Picturing the World (7th Edition)

The given expression.

Pre-Algebra Student Edition

log a =

Precalculus

For what values of a is continuous at every x?

University Calculus: Early Transcendentals (4th Edition)

Three cards are randomly selected, without replacement, from an ordinary deck of 52 playing cards. Compute the ...

A First Course in Probability (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.