/*! 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 48 Write each repeating decimal fir... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 48

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{027}=0.027027 \ldots$$

Short Answer

Expert verified
Answer: \(\frac{1}{37}\)

Step by step solution

01

Express the repeating decimal as a geometric series

Let's express the decimal number 0.027027... as a geometric series. To do this, we start by identifying the repeating part of the decimal, which is 027. Then, we break down the decimal into individual terms, each multiplied by the appropriate power of 0.001 since the repeating part has three decimal places. $$0.027027... = 0.027 * 10^{-0} + 0.027 * 10^{-3} + 0.027 * 10^{-6} + ...$$ Now, we have a geometric series with the first term (a) equal to 0.027 and the common ratio (r) equal to 10^{-3}.
02

Apply the formula for the sum of an infinite geometric series

For an infinite geometric series with a first term 'a' and a common ratio 'r', where |r| < 1, we can find the sum (S) of the series with the following formula: $$S = \frac{a}{1 - r}$$ In our case, a = 0.027 and r = 10^{-3}. Now, we can apply the formula to find the sum of the series: $$S = \frac{0.027}{1 - 10^{-3}}$$
03

Simplify the fraction to find the sum

Let's simplify the fraction to find the sum of the series: $$S = \frac{0.027}{1 - 10^{-3}} = \frac{0.027}{\frac{1000 - 1}{1000}} = \frac{0.027}{\frac{999}{1000}}$$ Now multiply both the numerator and denominator by 1000 to remove the fraction in the denominator: $$S = \frac{0.027 * 1000}{999}$$
04

Convert the ratio to integers

Finally, we want to express the sum as a ratio of two integers. To do this, multiply both the numerator and denominator by the appropriate power of 10 to remove the decimal point in the numerator: $$S = \frac{0.027 * 1000 * 10^3}{999 * 10^3} = \frac{27 * 10^3}{999 * 10^3} = \frac{27}{999}$$ Now, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (27): $$S = \frac{27}{999} = \frac{27 \div 27}{999 \div 27} = \frac{1}{37}$$ So, the repeating decimal 0.027027... can be expressed as a fraction \(\frac{1}{37}\).

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

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=2}^{\infty} \frac{(-1)^{k}}{\ln k}$$

Evaluate the limit of the following sequences. $$a_{n}=\int_{1}^{n} x^{-2} d x$$

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year \(1984 .\) Assume the CPI has increased by an average of \(3 \%\) per year since \(1984 .\) Let \(c_{n}\) be the CPI \(n\) years after \(1984,\) where \(c_{0}=100.\)

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{0}=5, n=0,1,2, \dots$$

An early limit Working in the early 1600 s, the mathematicians Wallis, Pascal, and Fermat were attempting to determine the area of the region under the curve \(y=x^{p}\) between \(x=0\) and \(x=1\) where \(p\) is a positive integer. Using arguments that predated the Fundamental Theorem of Calculus, they were able to prove that $$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1}\left(\frac{k}{n}\right)^{p}=\frac{1}{p+1}$$ Use what you know about Riemann sums and integrals to verify this limit.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

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.