/*! 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 75 Consider the following infinite ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 75

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty}(-1)^{k}$$

Short Answer

Expert verified
Question: Determine the first four terms of the alternating series $$\sum_{k=1}^{\infty}(-1)^{k}$$ and analyze if it has a limit. Answer: The first four terms of the series are -1, 1, -1, 1. The sequence of partial sums alternates between -1 and 0, and does not approach a single value as the number of terms increases. Therefore, the infinite series does not have a limit.

Step by step solution

01

Identify the sequence

The given infinite series is an alternating series: $$\sum_{k=1}^{\infty}(-1)^{k}$$
02

Find the first four terms of the sequence

The first four terms of the sequence can be found by replacing k with 1, 2, 3, and 4 in the formula. We get: Term 1: \((-1)^1 = -1\) \\ Term 2: \((-1)^2 = 1\) \\ Term 3: \((-1)^3 = -1\) \\ Term 4: \((-1)^4 = 1\) \\ So, the sequence looks like: -1, 1, -1, 1, ...
03

Calculate the first four partial sums of the sequence

To find the first four partial sums of the sequence, we will get the sum of the terms of the sequence up to the nth term. Partial Sum 1: \(S_1 = -1\) \\ Partial Sum 2: \(S_2 = -1 + 1 = 0\) \\ Partial Sum 3: \(S_3 = -1 + 1 - 1 = -1\) \\ Partial Sum 4: \(S_4 = -1 + 1 - 1 + 1 = 0\) \\ So, the first four partial sums are -1, 0, -1, 0.
04

Determine the limit of the sequence of partial sums

To determine the limit of the infinite series, we observe the behavior of the sequence of partial sums as the number of terms increases. The sequence of partial sums seems to alternate between -1 and 0. As n goes to infinity, the sequence does not approach a single value but continues to alternate between -1 and 0.
05

State the conclusion

The limit of the sequence of partial sums does not exist due to the alternating nature of the sequence. Therefore, the infinite series $$\sum_{k=1}^{\infty}(-1)^{k}$$ does not have a limit as well.

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 infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} \cos (\pi k)$$

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation \(f_{n+1}=f_{n}+f_{n-1}\), for \(n=1,2,3, \ldots,\) where \(f_{0}=0, f_{1}=1 .\) Each term of the sequence is the sum of its two predecessors. a. Write out the first ten terms of the sequence. b. Is the sequence bounded? c. Estimate or determine \(\varphi=\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}},\) the ratio of the successive terms of the sequence. Provide evidence that \(\varphi=(1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Verify the remarkable result that $$f_{n}=\frac{1}{\sqrt{5}}\left(\varphi^{n}-(-1)^{n} \varphi^{-n}\right)$$

Consider the geometric series $$S=\sum_{k=0}^{\infty} r^{k}$$ which has the value \(1 /(1-r)\) provided \(|r|<1 .\) Let \(S_{n}=\sum_{k=0}^{n-1} r^{k}=\frac{1-r^{n}}{1-r}\) be the sum of the first \(n\) terms. The remainder \(R_{n}\) is the error in approximating \(S\) by \(S_{n} .\) Show that $$R_{n}=\left|S-S_{n}\right|=\left|\frac{r^{n}}{1-r}\right|$$

Determine whether the following series converge or diverge. $$\sum_{k=2}^{\infty} \frac{4}{k \ln ^{2} k}$$

Evaluate the limit of the following sequences. $$a_{n}=\frac{75^{n-1}}{99^{n}}+\frac{5^{n} \sin n}{8^{n}}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.