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Chapter 9: Problem 51

State the Limit Comparison Test and give an example of its use.

Short Answer

Expert verified
The Limit Comparison Test states if two series are positive and the limit of their ratio as n goes to infinity is a positive finite number, then both series either converge or diverge. But in our example, the limit was not a positive finite number, so the test doesn't give any conclusion.

Step by step solution

01

Statement of the Limit Comparison Test

If two series \(a_n\) and \(b_n\) are positive for every term n, and the limit of their ratio as n approaches infinity is a positive finite number, i.e., \( 0 < L = \lim_{n \to \infty} \frac{a_n}{b_n} < \infty \), then either both series converge or both diverge.
02

Example of the Limit Comparison Test

Take the series \(a_n = \frac{1}{n}\) and \(b_n = \frac{1}{n^2}\). Calculate the limit of the ratio \(L = \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} n = \infty\). Since the limit is not a positive finite number, the test does not decide whether the series converges or diverges.

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Key Concepts

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

Convergence of Series
Understanding the convergence of an infinite series is crucial in calculus. A series is said to converge if the sum of its infinite terms approaches a specific finite value as the number of terms grows indefinitely. Imagine stacking an infinite number of progressively smaller blocks horizontally; if they can be arranged to fit within a specified length, we say the series of block lengths converges.

A common example is the geometric series. For instance, consider the series \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, ... \), where each term is half the previous one. The sum will never exceed 1, so we say the series converges to 1.
Divergence of Series
On the flip side, a series diverges if the sum of its terms doesn't stabilize or approach any particular number but instead grows indefinitely. A classic divergent series is the harmonic series \( \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, ... \). No matter how many terms we add, the sum keeps increasing and never settles on a finite number.

In the context of the limit comparison test, if the limit of the ratio of two series' terms isn't finite as the number of terms goes to infinity, it can't determine convergence. The harmonic series, because it doesn't bound itself to a finite sum, serves as a prototypical example of divergence.
Infinite Series
An infinite series is essentially an addition of infinitely many terms, denoted as \( a_1 + a_2 + a_3 + ... + a_n \), where n tends towards infinity. The behavior of these infinite sums, whether they converge or diverge, forms a large part of the study of series in calculus.

Understanding the nature of these series helps in many areas of science and mathematics, such as calculating the present value of annuities in finance or determining the behavior of electric circuits in engineering. The beauty of infinite series lies in their capacity to approximate complex functions or numbers with great precision, such as \( e \) or \( \pi \).
Calculus
Calculus, with its two main branches - differential and integral calculus - is the language of change and motion that mathematicians and scientists speak. It's used to find the slopes of curves, areas under curves, and many other limits that describe the real world.

The limit comparison test is one of the ways calculus helps us understand the behavior of infinite series, by comparing two series to make conclusions about their convergence or divergence. Whether we're computing the probability of an event, optimizing a process in business, or predicting the fluctuations of a physical system, calculus provides the tools to model and analyze these situations.

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Most popular questions from this chapter

Use power series to approximate the value of the integral with an error of less than 0.0001. (In Exercises 55 and 56 , assume that the integrand is defined as 1 when \(x=0\).) \(\int_{0}^{1 / 2} \frac{\arctan x}{x} d x\)

Approximate the sum of the series by using the first six terms.\(\sum_{n=0}^{\infty} \frac{(-1)^{n} 2}{n !}\)

Use the Integral Test to determine the convergence or divergence of the \(p\) -series. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\)

Probability A fair coin is tossed repeatedly. The probability that the first head occurs on the \(n\) th toss is given by \(P(n)=\left(\frac{1}{2}\right)^{n}\), where \(n \geq 1\) (a) Show that \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^{n}=1\). (b) The expected number of tosses required until the first head occurs in the experiment is given by \(\sum_{n=1}^{\infty} n\left(\frac{1}{2}\right)^{n}\) Is this series geometric? (c) Use a computer algebra system to find the sum in part (b).

Find a first-degree polynomial function \(P_{1}\) whose value and slope agree with the value and slope of \(f\) at \(x=c .\) Use a graphing utility to graph \(f\) and \(P_{1} .\) What is \(P_{1}\) called? $$ f(x)=\tan x, \quad c=\frac{\pi}{4} $$
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