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Chapter 8: Problem 1

Define sequence and give an example.

Short Answer

Expert verified
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 following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} 9(0.1)^{k}$$

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{6}=0.666 \dots$$

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{3}=0.333 \dots$$

Give an example of a bounded sequence that has a limit.

Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{1}{k^{2}+4}$$
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