Prove the limit below exists, by using the Binomial Theorem and the Monotone converge ONLY. Do not use power series, natural logs, or the properties of ?, as this limit must be established first to prove those subsequent properties and theorems. All bounds must be proven.
Prove the limit below exists, by using the Binomial Theorem and the Monotone converge ONLY. Do not use power series, natural logs, or the properties of ?, as this limit must be established first to prove those subsequent properties and theorems. All bounds must be proven.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Prove the limit below exists, by using the Binomial Theorem and the Monotone converge ONLY. Do not use power series, natural logs, or the properties of ?, as this limit must be established first to prove those subsequent properties and theorems. All bounds must be proven.
![The image contains a mathematical expression representing a limit, which is fundamental in calculus. The expression is as follows:
\[
\lim_{{n \to \infty}} \left(1 + \frac{1}{n}\right)^n
\]
This limit is crucial because it defines the mathematical constant \(e\), which is approximately equal to 2.71828. The constant \(e\) is the base of the natural logarithm and appears frequently in various fields of mathematics, including calculus and complex analysis.
### Explanation:
- The notation \(\lim_{{n \to \infty}}\) indicates that we are considering the limit as \(n\) approaches infinity.
- Inside the limit, we have the expression \(\left(1 + \frac{1}{n}\right)^n\), which increases as \(n\) increases.
As \(n\) becomes very large, \(\left(1 + \frac{1}{n}\right)^n\) approaches the constant \(e\).
Understanding this limit is fundamental in the study of exponential growth, compound interest, and many areas of higher mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F42eb3f21-115e-47f2-b59c-753292f0c3d6%2Fd131b103-862e-427b-8aee-eb0da230c4f7%2Ftoa9j9_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression representing a limit, which is fundamental in calculus. The expression is as follows:
\[
\lim_{{n \to \infty}} \left(1 + \frac{1}{n}\right)^n
\]
This limit is crucial because it defines the mathematical constant \(e\), which is approximately equal to 2.71828. The constant \(e\) is the base of the natural logarithm and appears frequently in various fields of mathematics, including calculus and complex analysis.
### Explanation:
- The notation \(\lim_{{n \to \infty}}\) indicates that we are considering the limit as \(n\) approaches infinity.
- Inside the limit, we have the expression \(\left(1 + \frac{1}{n}\right)^n\), which increases as \(n\) increases.
As \(n\) becomes very large, \(\left(1 + \frac{1}{n}\right)^n\) approaches the constant \(e\).
Understanding this limit is fundamental in the study of exponential growth, compound interest, and many areas of higher mathematics.
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