/*! 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 4 Determine whether the series con... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 18: Problem 4

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ \frac{2}{3}+2+6+\cdots+2(3)^{n}+\cdots $$

Short Answer

Expert verified
The given series diverges and hence does not have a finite sum.

Step by step solution

01

Identify the Pattern

The first step is identifying if the series follows a specific pattern. Looking at the given series \( \frac{2}{3}, 2, 6, \ldots \), the pattern can be seen as each term being three times the previous term after the first term. This indicates a geometric series with first term \(a = \frac{2}{3}\) and common ratio \(r = 3\).
02

Determine if the series Converges or Diverges

In a geometric series, a necessary condition for the series to converge is that the absolute value of the common ratio \(r\) is less than 1. However, in this case the common ratio is 3 which is greater than 1. Therefore, the given series will diverge.
03

Sum the Series

Calculating the sum for a geometric series only makes sense if the series converges. Since we've just determined this series diverges, its sum is not a finite number.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence and Divergence
When examining series, particularly infinite ones like geometric series, it's crucial to determine whether they converge or diverge. **Convergence** simply means that as we sum more and more terms from the series, the total approaches a specific finite value. **Divergence** implies that the sum does not settle; it might grow indefinitely or not stabilize at any finite number.
  • To test convergence in geometric series, the core factor is the common ratio, denoted as \( r \).
  • If the absolute value of \( r \) is less than 1, \( |r| < 1 \), the series converges.
  • Conversely, if \( |r| \geq 1 \), the series diverges.
In our original exercise, the common ratio \( r \) was 3, which led to divergence because it is greater than 1.
Common Ratio
The common ratio in a geometric series is a pivotal element that dictates the behavior of the series.Defined as the constant factor between successive terms, it is typically represented by \( r \).
  • For example, consider the series \( \frac{2}{3} , 2, 6, \ldots \).
  • Here, each term is calculated by multiplying the previous term by 3.
This multiplication by 3 establishes the common ratio as 3. To compute it in general, take any term in the series and divide it by the preceding term.If the terms are \( a, ar, ar^2, \ldots \), then the common ratio \( r = \frac{ar}{a} = r \).By understanding the common ratio, you can quickly assess whether a series will converge or diverge based on its value.
Infinite Series
An infinite series is a sum of an endless number of terms.This concept can be somewhat counterintuitive at first because although the number of terms seems boundless, the series might still have a finite sum if it converges.
  • Infinite series can be derived from patterns observed in sequences.
  • They expand endlessly in the form \( a, ar, ar^2, ar^3, \ldots \) for geometric types.
Understanding infinite series involves determining whether their total approaches a finite limit or not, as seen in the segment on convergence and divergence.In cases like the original exercise, where the series diverges, the infinite aspect means that no definite total sum exists and it stretches indefinitely.Through these insights, you can begin to see the interplay between finite concepts and infinite processes that create fascinating results in mathematics.

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

Determine whether the series converges or diverges. If it converges, \(n d\) its sum. $$ 2 e+2 e^{2}+2 e^{3}+\cdots+2 e^{n}+\cdots $$

Determine whether each of the following geometric series converges or diverges. If the series converges, determine to what it converges. (a) \(-\frac{4}{3}-\frac{1}{2}-\frac{3}{16}-\frac{9}{128}+\cdots\) (b) \(-\frac{1}{100}+\frac{1.1}{(100)^{2}}-\frac{1.21}{(100)^{3}}+\frac{1.331}{(100)^{4}}-\cdots\) (c) \(-\frac{7}{10000}+\frac{7}{11000}-\frac{7}{12100}+\frac{7}{13310}-\cdots\) (d) \(1-x+x^{2}-x^{3}+\cdots\) for \(|x|<1\)

People who have slow metabolism due to a malfunctioning thyroid can take thyroid medication to alleviate their condition. For example, the boxer Muhammad Ali took Thyrolar 3, which is 3 grains of thyroid medication, every day. The amount of the drug in the bloodstream decays exponentially with time. The half-life of Thyrolar is 1 week. (a) Suppose one 3 -grain pill of Thyrolar is taken. Write an equation for the amount of the drug in the bloodstream \(t\) days after it has been taken. (Hint: In part (a) you are dealing with one 3 -grain pill of Thyrolar. Knowing the half-life of Thyrolar, you are asked to come up with a decay equation. This part of the problem has nothing to do with geometric sums.) (b) Suppose that Ali starts with none of the drug in his bloodstream. If he takes 3 grains of Thyrolar every day for ve days, how much Thyrolar is in his bloodstream immediately after having taken the fth pill? (c) Suppose Ali takes 3 grains of Thyrolar each day for one month. How much thyroid medication will be in his bloodstream right before he takes his 31 st pill? Right after? (d) After taking this medicine for many years, what was the amount of the drug in his body immediately after taking a pill? Historical note: Before one of his last ghts Muhammad Ali decided to up his dosage to 6 grains. In doing so he mimicked the symptoms of an overactive thyroid. The result in terms of the ght was dismal.

Express each of the sums in closed form. Wherever possible, give a numerical approximation of the sum, rounded off to 3 decimal places. $$ \frac{2}{3}+\frac{1}{3}+\frac{1}{6}+\frac{1}{12}+\cdots+\frac{1}{3 \cdot 2^{100}} $$

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ p+p^{3}+p^{5}+p^{7}+p^{9} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.