/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Write out the first five terms o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 22

Write out the first five terms of the sequence, determine whether the sequence converges, and if so find its limit. $$ \left\\{\left(1-\frac{2}{n}\right)^{n}\right\\}_{n=1}^{+\infty} $$

Short Answer

Expert verified
The sequence is: \(-1, 0, \frac{1}{27}, \frac{1}{16}, \frac{243}{3125}\) and it converges to \( e^{-2} \).

Step by step solution

01

Calculate the first term

To find the first term of the sequence, set \( n = 1 \). Thus, the expression becomes \( \left(1 - \frac{2}{1}\right)^1 = (-1)^1 = -1 \).
02

Calculate the second term

For the second term, set \( n = 2 \). The expression then becomes \( \left(1 - \frac{2}{2}\right)^2 = \left(0\right)^2 = 0 \).
03

Calculate the third term

Set \( n = 3 \) for the third term. This gives \( \left(1 - \frac{2}{3}\right)^3 = \left(\frac{1}{3}\right)^3 = \frac{1}{27} \).
04

Calculate the fourth term

For the fourth term with \( n = 4 \), the expression becomes \( \left(1 - \frac{2}{4}\right)^4 = \left(\frac{1}{2}\right)^4 = \frac{1}{16} \).
05

Calculate the fifth term

For the fifth term, set \( n = 5 \). This results in \( \left(1 - \frac{2}{5}\right)^5 = \left(\frac{3}{5}\right)^5 = \frac{243}{3125} \).
06

Analyzing Convergence

As \( n \to \infty \), the term \( \frac{2}{n} \to 0 \), and the expression \( \left(1 - \frac{2}{n}\right)^n \) approaches \( e^{-2} \) due to the limit definition \( \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x \).
07

Conclusion of Convergence

Since \( \left(1 - \frac{2}{n}\right)^n \to e^{-2} \) as \( n \to \infty \), the sequence converges to \( e^{-2} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Limit of a Sequence
Understanding the limit of a sequence is crucial in determining if a sequence converges. In simple terms, the limit is the value a sequence approaches as the term number gets really large. Consider each term in the sequence like steps getting closer to a final point. This final point is the limit.
  • If a sequence gets closer to a specific finite number as the terms increase, it has a limit.
  • If terms only keep enlarging without settling around a number, the sequence has no limit.
  • In mathematical lingo, a sequence \( \{a_n\} \) converges to a limit \( L \) if \( \lim_{n \to \infty} a_n = L \).
Within our exercise, the sequence \( \left\{\left(1-\frac{2}{n}\right)^{n}\right\}_{n=1}^{+\infty} \) is shown to approach the number \( e^{-2} \). Therefore, \( e^{-2} \) is the limit of this sequence.
Sequence Convergence
Sequence convergence is about checking whether a sequence tends towards a particular number, known as its limit. It involves looking at the behavior of the sequence as we pick larger and larger terms.
  • A sequence converges if it approaches a single number when \( n \to \infty \).
  • If no such number exists, the sequence diverges.
  • Knowing how to determine convergence helps in analyzing these sequences mathematically.
Looking at our specific sequence, as \( n \) increases, \( \left(1-\frac{2}{n}\right)^{n} \) is influenced greatly by the property of exponential limits. It uses the principle that \( \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x \). Applying this rule here demonstrates convergence to \( e^{-2} \), a clear endpoint for our sequence.
Limit Definition in Calculus
In calculus, understanding limits is a massive part. Limits often help us explain changes and behavior of functions, including sequences, over time. This is fundamentally about understanding what happens as values get infinitely large or small.
  • It describes the value that a function or a sequence "approaches" as the input or index approaches some value.
  • For sequences, it helps in determining the behavior as the sequence gets to larger numbers.
  • Knowing how to leverage limits help in more complex calculus topics like derivatives and integrals.
In the context of this sequence, the limit definition used was \( \lim_{n \to \infty} \left(1 - \frac{2}{n}\right)^n \). By utilizing the identity for exponential functions in limits \( \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x \), calculating this becomes more intuitive.
The exercise sequence eventually levels to \( e^{-2} \), which perfectly falls within these rules and definitions found in calculus.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the radius of convergence and the interval of convergence. $$ \sum_{k=1}^{\infty}(-1)^{k+1} \frac{(x+1)^{k}}{k} $$

Determine whether the series converges. $$ \sum_{k=1}^{\infty} \frac{1}{\sqrt{k+5}} $$

Determine whether the statement is true or false. Explain your answer. The integral test can be used to prove that a series diverges.

Use the Remainder Estimation Theorem to find an interval containing \(x=0\) over which \(f(x)\) can be approximated by \(p(x)\) to three decimal-place accuracy throughout the interval. Check your answer by graphing \(|f(x)-p(x)|\) over the interval you obtained. $$ f(x)=\cos x ; p(x)=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !} $$

Use a graphing utility to confirm that the integral test applies to the series \(\sum_{k=1}^{\infty} k^{2} e^{-k},\) and then determine whether the series converges.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.