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Chapter 8: Problem 1

Define sequence and give an example.

Short Answer

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Question: Define what a sequence is and provide an example. Answer: A sequence is an ordered list of elements, in which each element is called a term. The terms in a sequence can be numbers, functions, or other mathematical objects, and can be either finite or infinite. An example of a sequence is the sequence of even numbers: {2, 4, 6, 8, ...}, which can be generated using the formula "a_n = 2n" for n = 1, 2, 3, ....

Step by step solution

01

Define a sequence

A sequence is an ordered list of elements, in which each element is called a term. The terms in a sequence can be numbers, functions, or other mathematical objects. Sequences can be finite or infinite, depending on the number of terms they contain. A sequence is usually denoted as (a_n) or {a_n} for n = 1, 2, 3, ..., where 'n' is the position of each element in the sequence, and 'a_n' represents the element at the nth position.
02

Provide an example

An example of a sequence would be the sequence of even numbers: {2, 4, 6, 8, ...}. In this sequence, the first term a_1 is 2, the second term a_2 is 4, the third term a_3 is 6, and so on. One could generate the terms of this sequence using a formula, such as "a_n = 2n" for n = 1, 2, 3, .... Using this formula, we can find any term of this sequence by substituting 'n' with the desired position. For example, to find the 5th term of this sequence, we can substitute n = 5 into the formula, and we get a_5 = 2脳5 = 10, so the 5th term in this sequence of even numbers is 10.

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Most popular questions from this chapter

Consider the sequence \(\left\\{x_{n}\right\\}\) defined for \(n=1,2,3, \ldots\) by $$x_{n}=\sum_{k=n+1}^{2 n} \frac{1}{k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2 n}.$$ a. Write out the terms \(x_{1}, x_{2}, x_{3}\). b. Show that \(\frac{1}{2} \leq x_{n}<1,\) for \(n=1,2,3, \ldots\). c. Show that \(x_{n}\) is the right Riemann sum for \(\int_{1}^{2} \frac{d x}{x}\) using \(n\) subintervals. d. Conclude that \(\lim _{n \rightarrow \infty} x_{n}=\ln 2\).

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\tan ^{-1}\left(\frac{10 n}{10 n+4}\right)$$

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. Jack took a \(200-\mathrm{mg}\) dose of a painkiller at midnight. Every hour, \(5 \%\) of the drug is washed out of his bloodstream. Let \(d_{n}\) be the amount of drug in Jack's blood \(n\) hours after the drug was taken, where \(d_{0}=200 \mathrm{mg}\)

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 magnitude of the remainder \(R_{n}\) is the error in approximating \(S\) by \(S_{n} .\) Show that $$ R_{n}=S-S_{n}=\frac{r^{n}}{1-r} $$

$$\text {Evaluate each series or state that it diverges.}$$ $$\sum_{k=1}^{\infty} \frac{(-2)^{k}}{3^{k+1}}$$
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