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Chapter 9: Problem 3

Explain how the Limit Comparison Test works.

Short Answer

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**Short answer question:** Explain the main steps of the Limit Comparison Test. **Answer:** The main steps of the Limit Comparison Test are: 1. Understand the given series and the Limit Comparison Test theorem. The theorem states that if the limit of the ratio of two series with positive terms is a positive finite number, either both series converge or both diverge. 2. Identify the given series (\(a_n\)) and another series (\(b_n\)) for comparison, ensuring that their terms are positive. 3. Compute the limit of the ratio of \(a_n\) and \(b_n\) as \(n\) approaches infinity. 4. Simplify the limit and determine its value. 5. Based on the theorem, if the limit is positive and finite, determine if both series either converge or diverge. If one series' convergence or divergence is known, the other series' behavior can be determined.

Step by step solution

01

Understand the series and the theorem

First, we must know that a series is an infinite sum of terms, often written in the form of \(sum_{n=1}^{infty} a_n\). The Limit Comparison Test is used to determine whether a series converges (has a finite sum) or diverges (has an infinite or undefined sum). Next, let's state the Limit Comparison Test theorem: Let \(sum_{n=1}^{infty} a_n\) and \(sum_{n=1}^{infty} b_n\) be two series with positive terms. If \(lim_{n rightarrow infty} frac{a_n}{b_n}=L\), where \(0<L<infty\), then either both series converge, or both series diverge.
02

Identify the given series

To use the Limit Comparison Test, we must first have a series \(sum_{n=1}^{infty} a_n\) and another series \(sum_{n=1}^{infty} b_n\). It is essential that the terms of both series are positive. The given exercise does not provide a specific series, so in the next step, we will take an example to illustrate the test.
03

Apply Limit Comparison Test to an example

Consider the series \(sum_{n=1}^{infty} frac{n}{n^2+1}\). Let this be our \(a_n\). To perform the Limit Comparison Test, we will compare this series with a known simpler series, like \(sum_{n=1}^{infty} frac{1}{n}\), which is the harmonic series and is known to diverge. Let this be our \(b_n\). Now let's compute the limit of the ratio of \(a_n\) and \(b_n\): $$lim_{n rightarrow infty} frac{a_n}{b_n} = lim_{n rightarrow infty} frac{frac{n}{n^2+1}}{frac{1}{n}}$$
04

Simplify the limit

To determine the limit, we can simplify the expression obtained in the previous step: $$lim_{n rightarrow infty} frac{frac{n}{n^2+1}}{frac{1}{n}} = lim_{n rightarrow infty} frac{n^2}{n^2+1}$$ Now, calculate the limit: $$lim_{n rightarrow infty} frac{n^2}{n^2+1} =frac{lim_{n rightarrow infty} n^2}{lim_{n rightarrow infty}(n^2+1)}$$ Since the degrees in the numerator and denominator are equal, the limit can be found by dividing the coefficients of the highest power terms, in this case, 1: $$frac{lim_{n rightarrow infty} n^2}{lim_{n rightarrow infty}(n^2+1)} =frac{1}{1} = 1 $$
05

Determine convergence or divergence

Since the limit of the ratio of \(a_n\) and \(b_n\) is 1, which is both positive and finite, the Limit Comparison Test tells us that either both series converge or both series diverge. We already know that the harmonic series (\(sum_{n=1}^{infty} frac{1}{n}\)) diverges. Therefore, the given series, \(sum_{n=1}^{infty} frac{n}{n^2+1}\), must also diverge. And that's how the Limit Comparison Test works. Refer to the steps above when explaining the test to a student, and ensure to use examples to illustrate the procedure.

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Most popular questions from this chapter

Infinite products An infinite product \(P=a_{1} a_{2} a_{3} \ldots,\) which is denoted \(\prod_{k=1}^{\infty} a_{k}\) is the limit of the sequence of partial products \(\left\\{a_{1}, a_{1} a_{2}, a_{1} a_{2} a_{3}, \dots\right\\}\) a. Show that the infinite product converges (which means its sequence of partial products converges) provided the series \(\sum_{k=1}^{\infty} \ln a_{k}\) converges. b. Consider the infinite product $$P=\prod_{k=2}^{\infty}\left(1-\frac{1}{k^{2}}\right)=\frac{3}{4} \cdot \frac{8}{9} \cdot \frac{15}{16} \cdot \frac{24}{25} \cdots$$ Write out the first few terms of the sequence of partial products, $$P_{n}=\prod_{k=2}^{n}\left(1-\frac{1}{k^{2}}\right)$$ (for example, \(P_{2}=\frac{3}{4}, P_{3}=\frac{2}{3}\) ). Write out enough terms to determine the value of the product, which is \(\lim _{n \rightarrow \infty} P_{n}\). c. Use the results of parts (a) and (b) to evaluate the series $$\sum_{k=2}^{\infty} \ln \left(1-\frac{1}{k^{2}}\right)$$

Pick two positive numbers \(a_{0}\) and \(b_{0}\) with \(a_{0}>b_{0}\) and write out the first few terms of the two sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}:\) $$a_{n+1}=\frac{a_{n}+b_{n}}{2}, \quad b_{n+1}=\sqrt{a_{n} b_{n}}, \quad \text { for } n=0,1,2 \dots$$ (Recall that the arithmetic mean \(A=(p+q) / 2\) and the geometric mean \(G=\sqrt{p q}\) of two positive numbers \(p\) and \(q\) satisfy \(A \geq G\). a. Show that \(a_{n}>b_{n}\) for all \(n\). b. Show that \(\left\\{a_{n}\right\\}\) is a decreasing sequence and \(\left\\{b_{n}\right\\}\) is an increasing sequence. c. Conclude that \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) converge. d. Show that \(a_{n+1}-b_{n+1}<\left(a_{n}-b_{n}\right) / 2\) and conclude that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) The common value of these limits is called the arithmetic-geometric mean of \(a_{0}\) and \(b_{0},\) denoted \(\mathrm{AGM}\left(a_{0}, b_{0}\right)\). e. Estimate AGM(12,20). Estimate Gauss' constant \(1 / \mathrm{AGM}(1, \sqrt{2})\).

Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\) ). a. \(\sum_{k=0}^{\infty} 0.72^{k}\) b. \(\sum_{k=0}^{\infty}(-0.25)^{k}\)

a. Sketch the function \(f(x)=1 / x\) on the interval \([1, n+1]\) where \(n\) is a positive integer. Use this graph to verify that $$ \ln (n+1)<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}<1+\ln n $$ b. Let \(S_{n}\) be the sum of the first \(n\) terms of the harmonic series, so part (a) says \(\ln (n+1)0,\) for \(n=1,2,3, \ldots\) c. Using a figure similar to that used in part (a), show that $$ \frac{1}{n+1}>\ln (n+2)-\ln (n+1) $$ d. Use parts (a) and (c) to show that \(\left\\{E_{n}\right\\}\) is an increasing sequence \(\left(E_{n+1}>E_{n}\right)\) e. Use part (a) to show that \(\left\\{E_{n}\right\\}\) is bounded above by 1 f. Conclude from parts (d) and (e) that \(\left\\{E_{n}\right\\}\) has a limit less than or equal to \(1 .\) This limit is known as Euler's constant and is denoted \(\gamma\) (the Greek lowercase gamma). g. By computing terms of \(\left\\{E_{n}\right\\},\) estimate the value of \(\gamma\) and compare it to the value \(\gamma \approx 0.5772 .\) (It has been conjectured, but not proved, that \(\gamma\) is irrational.) h. The preceding arguments show that the sum of the first \(n\) terms of the harmonic series satisfy \(S_{n} \approx 0.5772+\ln (n+1)\) How many terms must be summed for the sum to exceed \(10 ?\)

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. $$a_{n+1}=4 a_{n}\left(1-a_{n}\right) ; a_{0}=0.5, n=0,1,2, \dots$$
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