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Chapter 10: Problem 1

Explain how the Limit Comparison Test works.

Short Answer

Expert verified
Question: Explain the concept of the Limit Comparison Test and its applications in determining the convergence or divergence of two series, using an example to illustrate its application. Answer: The Limit Comparison Test is a method used to determine the convergence or divergence of two series by comparing their growth rates. If the limit of the ratio between the terms of the two series exists and is positive, it indicates that the two series either both converge or both diverge. This is because for large values of n, the terms of both series grow at a similar rate. An example of the Limit Comparison Test application is when comparing the series \(\sum_{n=1}^{\infty} \frac{n}{n^3+1}\) and \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). By calculating the limit of the ratio between their terms and evaluating it, we obtain a limit value of 1, which is positive. Since the series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) is a convergent p-series, we can conclude that \(\sum_{n=1}^{\infty} \frac{n}{n^3+1}\) also converges, based on the Limit Comparison Test.

Step by step solution

01

Define Limit Comparison Test

The Limit Comparison Test states that if you have two series, \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} b_n\), with \(a_n > 0\) and \(b_n > 0\) for all n, then you can take the limit as n approaches infinity of the ratio between their terms, \(a_n\) and \(b_n\). The limit is given by \(c = \lim_{n\to\infty} \frac{a_n}{b_n}\). If the limit, \(c\), exists and is positive, then both of the series will either both converge or both diverge.
02

Intuitive Explanation for the Test

The intuition behind this test is that if the limit of the ratio between the terms of the two series exists and is positive, this means that for large values of n, the terms of both series grow at roughly the same rate. Therefore, if one of the series converges or diverges, the other must as well, since their growth rates are similar.
03

Example of Limit Comparison Test Application

Let's consider an example of the Limit Comparison Test applied to \(\sum_{n=1}^{\infty} \frac{n}{n^3+1}\) and \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). We'll follow these steps: 1. Calculate the limit of the ratio between the terms. \(c=\lim_{n\to\infty} \frac{\frac{n}{n^3+1}}{\frac{1}{n^2}}\) 2. Simplify the limit expression. \(c=\lim_{n\to\infty} \frac{n^3}{n^3+1}\) 3. Evaluate the limit. \(c=1\) Since the limit, \(c\), exists and is positive, the two series either both converge or both diverge. Since we know that \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) is a convergent p-series with p = 2 (which satisfies the condition p > 1 for convergence), we can conclude that \(\sum_{n=1}^{\infty} \frac{n}{n^3+1}\) also converges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergence of Series
When we talk about convergence of series, we refer to the behavior of an infinite series as the number of terms increases indefinitely. A series is said to converge if the sequence of its partial sums approaches a finite limit. In simpler terms, if you keep adding terms to an infinite series and the total sum goes towards a specific number, rather than heading off to infinity, that series converges.

The importance of this concept rests on its ability to determine whether a series results in a finite value or not, making it foundational in understanding infinite series in mathematics. The Limit Comparison Test is one tool in a set of several tests used to determine the convergence or divergence of a series.
Infinite Series
An infinite series is a sum of infinitely many terms, laid out as \( \sum_{n=1}^\infty a_n \), where \( a_n \) represents the nth term in the series. We often encounter such series in calculus and mathematical analysis.

The fascinating bit about these series is not just their infinity, but the fact they can oftentimes sum up to specific, finite numbers. Analysis of such series can inform understanding of patterns within sequences and functions. Classifying them, which involves determining whether a given series has a definite sum (converges) or not (diverges), is an essential skill for studying advanced mathematics.
P-Series
The p-series is a particular kind of infinite series given in the form of \( \sum_{n=1}^\infty \frac{1}{n^p} \), with \( p \) being a positive real number. The convergence of a p-series depends on the value of \( p \). It converges if \( p > 1 \) and diverges for \( p \leq 1 \). Understanding the behavior of p-series is essential as they are often used as a comparison series to ascertain the convergence or divergence of other more complex series, utilizing tests like the Limit Comparison Test.

The simple form of a p-series makes it a convenient tool for comparison. For example, if a series behaves similarly to a known convergent p-series for large values of \( n \) (as the terms approach infinity), it can often be inferred that the series in question also converges.
Limits in Calculus
In calculus, limits are used to describe the value that a function or sequence 'approaches' as the input or index approaches some value. Limits are foundational for defining continuity, derivatives, and integrals鈥攚hich are the core concepts of calculus.

With the Limit Comparison Test, limits help to compare the terms of two series as \( n \) goes to infinity to determine if both series exhibit similar behavior. Essentially, the limit tells us about the end behavior of a sequence, and consequently, about the convergence or divergence of an infinite series, based on how the terms of the series behave 'at infinity'.

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Most popular questions from this chapter

Suppose an alternating series with terms that are nonincreasing in magnitude converges to a value \(L\). Explain how to estimate the remainder that occurs when the series is terminated after \(n\) terms.

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 \(f_{0}=1\) and \(f_{1}=1 .\) 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 \(\varphi=\frac{1+\sqrt{5}}{2},\) a number known as the golden mean. d. Use induction to verify the remarkable result that \(f_{n}=\frac{1}{\sqrt{5}}\left(\varphi^{n}-(-1)^{n} \varphi^{-n}\right)\)

Remainder term Consider the geometric series \(S=\sum_{k=0}^{\infty} r^{k},\) which has the value \(\frac{1}{1-r}\) 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 magnitude of the remainder \(R_{n}\) is the error in approximating \(S\) by \(S_{n}\). Show that $$R_{n}=S-S_{n}=\frac{r^{n}}{1-r}$$

Explicit formulas Write the first four terms of the sequence \(\left {a_{n}\right\\}_{n=1}^{\infty}\) $$a_{n}=\frac{1}{10^{n}}$$

Fixed-point iteration A method for estimating a solution to the equation \(x=f(x)\). known as fixed-point iteration, is based on the following recurrence relation. Let \(x_{0}=c\) and \(x_{n+1}=f\left(x_{n}\right)\) for \(n=1,2,3, \ldots\) and a real number \(c .\) lf the sequence \(\left\\{x_{n}\right\\}_{n=0}^{\infty}\) converges to \(L\), then \(L\) is a solution to the equation \(x=f(x)\) and \(L\) is called a fixed point of \(f .\) To estimate \(L\) with \(p\) digits of accuracy to the right of the decimal point, we can compute the terms of the sequence \(\left\\{x_{n}\right\\}_{n=0}^{\infty}\) until two successive values agree to \(p\) digits of accuracy. Use fixed-point iteration to find a solution to the following equations with \(p=3\) digits of accuracy using the given value of \(x_{0}\) $$x=\frac{\sqrt{x^{3}+1}}{20} ; x_{0}=5$$
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