/*! 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} Problem 22 Use the Ratio Test to determine ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 22

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} n\left(\frac{3}{2}\right)^{n} $$

Short Answer

Expert verified
The infinite series \( \sum_{n=1}^{\infty} n\left(\frac{3}{2}\right)^{n} \) diverges.

Step by step solution

01

Identify the terms of the series

Firstly, identify the terms of the series. In this case, the term \( a_n \) is given by \( n (\frac{3}{2})^n \) where \( n \) ranges from 1 to infinity.
02

Calculate the ratio of the consecutive terms

Next, calculate the ratio \( \frac{a_{n+1}}{a_n} \) which will transform to \( \frac{(n+1)(\frac{3}{2})^{n+1}}{n (\frac{3}{2})^n} \) . When simplified this becomes \( \frac{3}{2} \cdot \frac{n+1}{n} \) .
03

Find the limit

To apply the Ratio Test, find the limit of the ratio as \( n \) approaches infinity. Take the limit as \( n \rightarrow \infty \) of \( \frac{3}{2} \cdot \frac{n+1}{n} \) . This results in \( \frac{3}{2} \) .
04

Use the Ratio Test to determine convergence or divergence

Finally determine whether the series converges or diverges. According to the Ratio Test, if the limit is less than 1, the series converges. If the limit is more than 1, the series diverges. If the limit equals 1, it is inconclusive. Since the limit in this case is \( \frac{3}{2} \), which is more than 1, this means the series diverges.

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

Verify that the geometric series converges. $$ \sum_{n=0}^{\infty}(-0.6)^{n}=1-0.6+0.36-0.216+\cdots $$

Find the sum of the convergent series. $$ \sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}-\frac{1}{3^{n}}\right) $$

Verify that the geometric series converges. $$ \sum_{n=0}^{\infty}(0.9)^{n}=1+0.9+0.81+0.729+\cdots $$

Probability: Coin Toss A fair coin is tossed until a head appears. The probability that the first head appears on the \(n\) th toss is given by \(P=\left(\frac{1}{2}\right)^{n},\) where \(n \geq 1 .\) Show that $$\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=1$$

Determine the convergence or divergence of the series. Use a symbolic algebra utility to verify your result. $$ \sum_{n=0}^{\infty} \frac{3}{4^{n}} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Decision Maths

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.