/*! 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 26 Find the sum of the convergent s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 26

Find the sum of the convergent series. $$ 4-2+1-\frac{1}{2}+\cdots $$

Short Answer

Expert verified
The sum of the series is \( \frac{8}{3} \).

Step by step solution

01

Identify the first term and the common ratio

The first term \(a\) is 4 and the common ratio \(r\) can be obtained by dividing any term by its preceding term. As a result, the ratio \(r\) is \( -1/2 \).
02

Apply the formula for the sum of an infinite geometric series

Substitute the values of \(a\) and \(r\) into the formula \(S = \frac{a}{1 - r}\), yielding \(S = \frac{4}{1 - (-1/2)}\).
03

Simplify the sum

Solve the expression to get the sum \(S\), this gives us \(S = \frac{4}{1 + 1/2} = \frac{4}{3/2} = 4 \cdot \frac{2}{3} = \frac{8}{3}\).

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

Budget Analysis A government program that currently costs taxpayers 1.3 billion dollars per year is to be cut back by \(15 \%\) per year. (a) Write an expression for the amount budgeted for this program after \(n\) years. (b) Compute the budget amounts for the first 4 years. (c) Determine the convergence or divergence of the sequence of reduced budgets. If the sequence converges, find its limit.

Biology Suppose that you have a single bacterium able to divide to form two new cells every half hour. At the end of the first half hour there are two individuals, at the end the first hour there are are two individuals, at the end of the first hour there are four individuals, and so on. (a) Write an expression for the \(n\) th term of the sequence. (b) How many bacteria will there be after 10 hours? After 20 hours? (Source: Adapted from Levine/Miller, Biology: Discovering Life, Second Edition)

Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) $$ 2,-4,6,-8,10, \dots $$

Annuity A deposit of 100 dollars is made at the beginning of each month for 5 years in an account that pays \(10 \%\) interest, compounded monthly. Use a symbolic algebra utility to find the balance \(A\) in the account at the end of the 5 years. $$A=100\left(1+\frac{0.10}{12}\right)+\cdots+100\left(1+\frac{0.10}{12}\right)^{60}$$

Investment A deposit of 100 dollars is made each month in an account that earns \(6 \%\) interest, compounded monthly. The balance in the account after \(n\) months is given by \(A_{n}=100(201)\left[(1.005)^{n}-1\right] .\) (a) Compute the first six terms of this sequence. (b) Find the balance after 5 years by computing the 60 th term of the sequence. (c) Find the balance after 20 years by computing the 240 th term of the sequence.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.