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Chapter 10: Problem 1

Give an example of a nonincreasing sequence with a limit.

Short Answer

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Question: Provide an example of a nonincreasing sequence with a limit and identify the limit. Answer: An example of a nonincreasing sequence with a limit is the sequence {1, 1/2, 1/3, 1/4, ...} with a(n) = 1/n for n = 1, 2, 3, ... . The limit of this sequence is 0 as n approaches infinity.

Step by step solution

01

Choose a nonincreasing sequence

Let's choose the sequence {1, 1/2, 1/3, 1/4, ...}, which can be represented by the function a(n) = 1/n for n = 1, 2, 3, ... . We can see that each term is less than or equal to the previous one as the denominators are increasing, so it is a nonincreasing sequence.
02

Find the limit of the sequence

To find the limit of the sequence, we need to identify the value the sequence is approaching as n goes to infinity. So, we need to examine the function a(n) = 1/n as n approaches infinity. \(\lim\limits_{n\to\infty} \frac{1}{n}\) As n approaches infinity, the value of 1/n approaches 0. Therefore, the limit of the sequence is 0.
03

Provide the example

The nonincreasing sequence {1, 1/2, 1/3, 1/4, ...} with a(n) = 1/n for n = 1, 2, 3, ... is an example of a nonincreasing sequence with a limit. The limit of this sequence is 0 as n approaches infinity.

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Key Concepts

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

Sequence Limit
Understanding the concept of a **sequence limit** is crucial in calculus and higher-level math. A sequence is a list of numbers in a specific order. The **limit of a sequence** refers to the value that the terms of the sequence get closer to as the sequence progresses indefinitely. For instance, in our exercise, we examined the sequence \( a(n) = \frac{1}{n} \).
The terms \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \) get smaller and approach the value of 0. As \( n \) becomes larger and larger, the terms \( \frac{1}{n} \) become so small that they essentially reach 0. Hence, in mathematical terms, we say:
\[ \lim\limits_{n\to\infty} \frac{1}{n} = 0 \]
  • Nonincreasing Condition: Each term is the same or less than the previous one.
This characteristic ensures the sequence converges to a definite limit, making it a vital study area in calculus.
Infinite Sequence
**Infinite sequences** are collections of numbers listed in a specific order that continue indefinitely. Represented by functions like \( a(n) = \frac{1}{n} \) in our example, a key trait of these sequences is their progression towards a limit.
In our case, the sequence \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \) is not only nonincreasing but also infinite. It goes on without ending.
  • Infinite Length: The sequence does not stop. It progresses as long as we want it to.
  • Approaching Limit: Even though the numbers keep changing, they hone in on a certain value, in this case, 0.
Understanding infinite sequences helps in perceiving how numbers behave when extended beyond our usual finite limits. In calculus, exploring the behavior of these sequences helps us understand limits, continuity, and other profound concepts.
Calculus Example
This scenario of a **calculus example** deals with finding limits and understanding sequences. In calculus, sequences like \( a(n) = \frac{1}{n} \) often serve as simple yet powerful teaching tools.
Calculus employs such sequences to illustrate the principles of limits and convergence.
  • Real-world Application: Limits help us model behaviors in physics and engineering, such as understanding gradual changes in velocity or population growth.
  • Sequence Behavior: Observing how \( \frac{1}{n} \) diminishes to 0 helps grasp concepts of convergence.
Addressing such basic examples builds a foundation, preparing students to tackle more complex problems involving integrals and derivatives. Such exercises not only solidify understanding but also demonstrate the interconnected nature of mathematical principles.

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Most popular questions from this chapter

Periodic savings Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of \(r\), which is the annual interest rate divided by 12 (for example, if the annual interest rate is \(2.4 \%, r=0.024 / 12=0.002) .\) For an initial investment of \(m\) dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or \(m+m(1+r) .\) Continuing in this fashion, it can be shown that the amount of money in your account after \(n\) months is \(A_{n}=m+m(1+r)+\cdots+m(1+r)^{n-1} .\) Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate. Monthly deposits of 250 dollars at a monthly interest rate of \(0.2 \%\)

Determine whether the following statements are true and give an explanation or counterexample. a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series. b. The series \(\sum_{k=3}^{\infty} \frac{1}{k \ln ^{p} k}\) converges for the same values of \(p\) as the series \(\sum_{k=3}^{\infty} \frac{1}{k^{p}}\) c. Both the Ratio Test and the Root Test can be applied conclusively to a geometric series. d. The Alternating Series Test can be used to show that some series diverge.

Determine whether the following series converge. Justify your answers. $$\sum_{k=3}^{\infty} \frac{5}{2+\ln k}$$

The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge. $$\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{10 k^{3}+k}{9 k^{3}+k+1}\right)^{k}$$

Explain why, with a series of positive terms, the sequence of partial sums is an increasing sequence.
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