Question

Determine whether the sequence converges or diverges. If convergent, give the limit of the sequence. $$\left\\{a_{n}\right\\}=\left\\{\left(1+\frac{1}{n}\right)^{n}\right\\}$$

Short answer

The sequence converges to \( e \).

Step-by-step solution

Identify the Sequence

Our sequence is \( \{a_n\} = \left( 1 + \frac{1}{n} \right)^n \). We need to determine if this sequence converges or diverges as \( n \) approaches infinity.

Recognize the Limit

The sequence \( \left( 1 + \frac{1}{n} \right)^n \) is known to approach a well-known mathematical constant as \( n \to \infty \). This limit is the base of the natural logarithm, \( e \).

Apply the Exponential Limit Definition

By the definition and exploration in calculus, \( \left( 1 + \frac{1}{n} \right)^n \to e \) as \( n \to \infty \). This result can be shown rigorously using the limit definition of \( e \) from calculus.

Confirm the Limit is Finite

Since the sequence converges to \( e \), which is a finite number, we can conclude that the sequence \( \{a_n\} \) converges.

Key concepts

Limit of a Sequence

A sequence is essentially a list of numbers that follow a certain pattern. Mathematicians often want to know if these numbers approach a specific value as they extend infinitely. This target value is called the 'limit' of the sequence. In our exercise, we have the sequence \( \left\{ a_{n} \right\} = \left( 1 + \frac{1}{n} \right)^{n} \). The question is whether or not this sequence has a limit as \( n \to \infty \).

When we look for a limit, we evaluate what happens to \( a_n \) as \( n \) becomes very large. A sequence either converges (reaches a limit) or diverges (doesn't approach a particular value). By analyzing patterns and using mathematical definitions, we determine if a sequence settles towards a single number, which indicates convergence.

The given sequence is known to converge to a constant value, which mathematicians have identified as "\( e \)." Therefore, the limit of our sequence is \( e \), and the sequence is convergent.

Natural Exponential Function

The natural exponential function is often represented as \( e^x \), where \( e \) is a mysterious yet fundamental number in mathematics. The value of \( e \) is approximately 2.71828. Despite looking like just another number, \( e \) emerges naturally in many areas of mathematics, particularly in growth processes and financial calculations.

Why is \( e \) so special? It's because of its unique properties related to growth and decay. When processes involve continuous and compound growth, \( e \) often appears as the base for the exponential function \( e^x \). For instance, when interest in a bank account is compounded continuously, the formula for calculating the balance uses \( e \).

• \( e \) is called the "natural" exponential base because it arises naturally in problems that involve exponential growth.
• It’s used to calculate continuous growth, which means that it's key in fields like finance, biology, and physics.

Understanding \( e \) allows us to model real-world phenomena where changes happen continuously, not just at fixed intervals.

Calculus Definition of 'e'

The number \( e \) is not just a random constant, but rather an important foundational element in calculus. Its definition is tied to the way functions change smoothly. In calculus, \( e \) is formally defined using the limit of \( \left( 1 + \frac{1}{n} \right)^{n} \) as \( n \to \infty \).

This definition from calculus shows the relationship between \( e \) and limits. When you observe how functions behave as they stretch to infinity, \( e \) naturally emerges because it's associated with processes that change at a constant rate.

• It's the limit to which the sequence \( \left( 1 + \frac{1}{n} \right)^{n} \) converges, demonstrating the innate link between \( e \) and exponential functions.
• This relationship is crucial for understanding concepts like derivatives and integrals, foundational tools for examining change and area in functions.

Grasping the calculus definition of \( e \) shows why it's involved in exponential growth and decay processes—it represents the most efficient rate of continuous change.