/*! 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 18 Evaluate \(\sum_{m=3}^{\infty} \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 18

Evaluate \(\sum_{m=3}^{\infty} \frac{8}{3^{m}}\).

Short Answer

Expert verified
The sum of the infinite geometric series, \(\sum_{m=3}^{\infty} \frac{8}{3^{m}}\), is \(\frac{4}{9}\).

Step by step solution

01

Identify the first term and common ratio #

The given series is: \[\sum_{m=3}^{\infty} \frac{8}{3^{m}}\] We can rewrite the series as follows: \[\sum_{n=0}^{\infty} \frac{8}{3^{n+3}}\] Now we can identify the first term (a) and the common ratio (r): a = \(\frac{8}{3^{0+3}}\) = \(\frac{8}{27}\) r = \(\frac{1}{3}\)
02

Apply the formula for the sum of an infinite geometric series #

The general formula for the sum of an infinite geometric series is: \[S = \frac{a}{1-r}\] Substituting the values of a and r into the formula, we get: \[S = \frac{\frac{8}{27}}{1-\frac{1}{3}}\]
03

Simplify the expression #

Next, simplify the denominator: \[1-\frac{1}{3} = \frac{2}{3}\] Now, divide the numerator by the simplified denominator: \[S = \frac{\frac{8}{27}}{\frac{2}{3}}\] \(\frac{(\frac{8}{27})}{(\frac{2}{3})}\) simplifies to: \[S = \frac{8}{27} \times \frac{3}{2}\] \[S = \frac{8}{18}\] \[S = \frac{4}{9}\]
04

Final Answer #

The sum of the given infinite geometric series is: \[S = \frac{4}{9}\]

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

Evaluate \(\lim _{n \rightarrow \infty} n \ln \left(1+\frac{1}{n}\right)\).

Consider the fable from the beginning of Section 3.4. In this fable, one grain of rice is placed on the first square of a chessboard, then two grains on the second square, then four grains on the third square, and so on, doubling the number of grains placed on each square. Find the smallest number \(n\) such that the total number of grains of rice on the first \(n\) squares of the chessboard is more than 30,000,000 .

Use Pascal's triangle to simplify the indicated expression. $$ (2-\sqrt{3})^{5} $$

Show that if \(|r|<1,\) then $$ \sum_{m=1}^{\infty} r^{m}=\frac{r}{1-r} $$

Evaluate the geometric series. $$ \sum_{m=5}^{91}(-2)^{m} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure 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.