/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q. 58 In Exercises 55鈥� 58 use the ra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q. 58 (page 592)

In Exercises 55鈥� 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

k!22k!

Short Answer

Expert verified

The given sequence is strictly decreasing fork鈮�1.

Step by step solution

01

Step 1. Given Information.

The given sequence isk!22k!.

02

Step 2. Use the ratio test. 

To analyze the monotonicity of the given sequence we will use the ratio test.

Let the general term of the sequence is ak=k!22k!.

So, the term ak+1is

ak+1=k+1!22k+1!.

According to the ratio test,

localid="1649233296833" ak+1ak=k+1!22k+1!k!22k!=k+1!k+1!2k+2!k!k!2k!=k+1k!k+1k!2k+22k+12k!k!k!2k!=k+1k+12k+12k+1=k+122k+1=32k+2

Now, ak+1ak<1fork鈮�1.

Thus, the given sequence is strictly decreasing fork鈮�1.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An Improper Integral and Infinite Series: Sketch the function f(x)=1xfor x 鈮� 1 together with the graph of the terms of the series 鈭�k=1鈭�1k.Argue that for every term Snof the sequence of partial sums for this series,Sn>鈭�1n+11xdx. What does this result tell you about the convergence of the series?

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

鈭�k=0鈭�鈥�e鈭�k鈭�k=0鈭�鈥�e鈭�k

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder, R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that localid="1649224052075" Rn鈮�10-6.

鈭�k=0鈭�e-k

Improper Integrals: Determine whether the following improper integrals converge or diverge.

鈭�1鈭�1xdx.

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder,R10.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so thatRn鈮�10-6

鈭�k=1鈭�1k2
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.