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Chapter 8: Problem 4

Give an example of a bounded sequence without a limit.

Short Answer

Expert verified
Question: Provide an example of a sequence that is bounded but does not converge to any limit. Answer: An example of such a sequence is a_n = (-1)^n, where n is a natural number. This sequence is bounded between -1 and 1 but does not converge to any limit as it oscillates between the two values indefinitely.

Step by step solution

01

Example of a bounded sequence without a limit

One such example is the sequence (-1)^n, where n is a natural number. Step by step breakdown of the sequence:
02

Define the sequence

Define a sequence a_n = (-1)^n, where n is a natural number (n = 1, 2, 3, ...).
03

Analyze the sequence pattern

Observe how the terms in the sequence alternate as the value of n increases: a_1 = (-1)^1 = -1, a_2 = (-1)^2 = 1, a_3 = (-1)^3 = -1, a_4 = (-1)^4 = 1, and so on.
04

Determine if the sequence is bounded

Since the sequence alternates between -1 and 1, it is clear that the sequence is bounded. We can express this by saying: -1 鈮� a_n 鈮� 1, for all natural numbers n.
05

Determine if the sequence has a limit

The sequence does not converge to a single value, as it continues to oscillate between -1 and 1 indefinitely. Therefore, the sequence does not have a limit.

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Most popular questions from this chapter

James begins a savings plan in which he deposits \(\$ 100\) at the beginning of each month into an account that earns \(9 \%\) interest annually or, equivalently, \(0.75 \%\) per month. To be clear, on the first day of each month, the bank adds \(0.75 \%\) of the current balance as interest, and then James deposits \(\$ 100\). Let \(B_{n}\) be the balance in the account after the \(n\) th deposit, where \(B_{0}=\$ 0\). a. Write the first five terms of the sequence \(\left\\{B_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{B_{n}\right\\}\). c. How many months are needed to reach a balance of \(\$ 5000 ?\)

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