/*! 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 5 Define infinite series and give ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 5

Define infinite series and give an example.

Short Answer

Expert verified
Answer: An infinite series is the sum of the terms of an infinite sequence, represented as \(\sum_{n=1}^{\infty} a_n\). To calculate the sum of an infinite geometric series, use the formula \(S = \frac{a}{1-r}\), where S is the sum, a is the first term, and r is the common ratio. Note that not all infinite series have a finite sum, as some may diverge.

Step by step solution

01

1. Define Infinite Series

An infinite series is the sum of the terms of an infinite sequence. In other words, it is the result of adding up an infinite number of individual elements or terms. It can be represented as: \[ \sum_{n=1}^{\infty} a_n \] where \(a_n\) is the n-th term of the sequence. Keep in mind that not all infinite series have a finite sum; some of them may diverge, which means their sum approaches infinity or does not settle to a fixed value.
02

2. Example of an Infinite Series

The most common example of an infinite series comes from the geometric series. Let's use the following geometric series as an example: \[ \sum_{n=1}^{\infty} ar^{n-1} \] where, a is the first term of the series, r is the common ratio, and n is the position of the term in the series. For our example, let's take \(a = 1\) and \(r = \frac{1}{2}\).
03

3. Write Out the Terms of the Series

Now, let's write out the terms of our example series and get a better understanding of it: \[ S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \] This infinite series is the sum of terms that keep on getting halved. We can notice that this series will get closer and closer to 2 but never actually reach it, as it's an infinite series.
04

4. Compute the Sum of the Infinite Series

To find the sum of an infinite geometric series, we use the formula: \[ S = \frac{a}{1-r} \] where S is the sum, a is the first term, and r is the common ratio. In our example, a = 1 and r = 1/2. Plugging these values into the formula: \[S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2\] So, the sum of our example infinite series is 2. In conclusion, we have defined an infinite series as the sum of the terms of an infinite sequence and used the geometric series as an example. We wrote out its terms and found its sum using the corresponding formula. Our example infinite series converges to a finite sum of 2.

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

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. $$a_{n+1}=\sqrt{2+a_{n}} ; a_{0}=1, n=0,1,2, \dots$$

Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty} x^{k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

a. Consider the number 0.555555...., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 \ldots\) b. Consider the number \(0.54545454 \ldots,\) which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots \ldots, n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form of the number. d. Try the method of part (c) on the number \(0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\frac{\ln (1 / n)}{n}\right\\}$$

Consider the following convergent series. a. Find an upper bound for the remainder in terms of \(n\) b. Find how many terms are needed to ensure that the remainder is less than \(10^{-3}\) c. Find lower and upper bounds $$\left(L_{n} \text { and } U_{n}\right.$$ respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. $$\sum_{k=1}^{\infty} \frac{1}{k^{8}}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

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.