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Chapter 11: Problem 1

(a) What is the difference between a sequence and a series? (b) What is a convergent series? What is a divergent series?

Short Answer

Expert verified
Sequences list numbers, series sum them. Convergent series approach a limit; divergent series do not.

Step by step solution

01

Understanding a Sequence

A sequence is an ordered list of numbers, usually defined by a specific rule or pattern. For example, the sequence \( a_1, a_2, a_3, \ldots \) represents the terms and is typically determined by a formula like \( a_n = n^2 \), which gives the sequence \( 1, 4, 9, 16, \ldots \). Each element in the sequence is identified by a position, known as the index.
02

Understanding a Series

A series is a sum of the terms of a sequence. For example, considering the sequence \( a_1, a_2, a_3, \ldots \), the series would be \( S_n = a_1 + a_2 + a_3 + \ldots + a_n \). It can be finite or infinite, depending on the context. For instance, the series derived from the sequence \( n^2 \) would be \( 1 + 4 + 9 + 16 + \ldots \).
03

Definition of a Convergent Series

A convergent series is an infinite series in which the sum approaches a specific number, called the limit, as more terms are added. Mathematically, if \( S_n \) is the partial sum of the first \( n \) terms of the series, the series is convergent if \( \lim_{n \to \infty} S_n \) exists and is finite.
04

Definition of a Divergent Series

A divergent series is an infinite series that does not converge to a limit. This means that as you add more terms, the sum \( S_n \) does not approach any finite value. Examples of divergent series include those where the sum tends to infinity or oscillates between different values.

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Key Concepts

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

Convergent Series
A convergent series is one of the more manageable types of infinite series. In mathematics, it represents an infinite series where the sum of its terms approaches a fixed number. This fixed number is known as the "limit" of the series. Think of it like trying to fill a cup with water, where the water droplets keep getting smaller and smaller until eventually, the cup is full. For a series to be convergent, the sequence of partial sums must have a finite limit.

Here are some key points about convergent series:
  • A series \( S \) of the form \( S = a_1 + a_2 + a_3 + \ldots \) is convergent if \( \lim_{n \to \infty} S_n \) exists and is finite.
  • In simple terms, the sequence of partial sums \( S_n \), which are sums of the first \( n \) terms, must get closer and closer to a specific number.
  • If after adding countless numbers, you find that total stabilizes (doesn鈥檛 keep increasing or decreasing indefinitely), you have a convergent series.
Convergent series are significant in various fields, including calculus and physics, because they provide predictable results.
Divergent Series
Intriguingly different from convergent series, divergent series do not settle at a limit. Imagine trying to measure the length of a road that keeps getting longer. That鈥檚 like a divergent series. Essentially, as more terms are added, the series doesn鈥檛 approach a specific endpoint.

Here are some defining traits of divergent series:
  • A series \( S = a_1 + a_2 + a_3 + \ldots \) is divergent if the limit \( \lim_{n \to \infty} S_n \) does not exist or is infinite.
  • As you keep adding more terms in a divergent series, the sum becomes either very large, heads towards infinity, or keeps bouncing around without settling on a single value.
  • Divergent series can behave in unpredictable ways, making them interesting yet challenging to work with.
Divergent series are important in theoretical discussions and can appear in real-life phenomena, despite their arithmetic unpredictability.
Infinite Series
Infinite series are like a never-ending story in mathematics. They are formed by adding terms of a sequence without a fixed endpoint. Unlike finite series, which have a defined number of terms to sum, infinite series continue indefinitely.

Some important aspects of infinite series include:
  • The series \( S = a_1 + a_2 + a_3 + \ldots \) has no end since it comprises an infinite number of terms.
  • Infinite series can be categorized based on their behavior: convergent (if they reach a limit) or divergent (if they do not reach a limit).
  • Infinite series are central in many areas of mathematics and science, providing continuous models in fields like economics and physics.
Understanding whether an infinite series converges or diverges is crucial, as it determines the kind of mathematical tool it becomes in problem-solving.

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Most popular questions from this chapter

If \(\Sigma a_{n}\) and \(\Sigma b_{n}\) are both divergent, is \(\Sigma\left(a_{n}+b_{n}\right)\) necessarily divergent?

Let \(a\) and \(b\) be positive numbers with \(a > b .\) Let \(a_{1}\) be their arithmetic mean and \(b_{1}\) their geometric mean: $$ a_{1}=\frac{a+b}{2} \quad b_{1}=\sqrt{a b} $$ Repeat this process so that, in general, $$ a_{n+1}=\frac{a_{n}+b_{n}}{2} \quad b_{n+1}=\sqrt{a_{n} b_{n}} $$ (a) Use mathematical induction to show that $$ a_{n}>a_{n+1}>b_{n+1}>b_{n} $$ (b) Deduce that both \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are convergent. (c) Show that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) Gauss called the common value of these limits the arithmetic-geometric mean of the numbers a and \(b\) .

A car is moving with speed 20 \(\mathrm{m} / \mathrm{s}\) and acceleration 2 \(\mathrm{m} / \mathrm{s}^{2}\) a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?

Find the sum of the series. $$\sum_{n=0}^{\infty} \frac{(-1)^{n} \pi^{2 n+1}}{4^{2 n+1}(2 n+1) !}$$

Test the series for convergence or divergence. $$\sum_{k=1}^{\infty} \frac{k \ln k}{(k+1)^{3}}$$
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