Question

Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. $$ a_{n}=\left(1-\frac{2}{n}\right)^{n} $$

Short answer

The sequence converges to \( \frac{1}{e^2} \).

Step-by-step solution

Analyze the Sequence

We have the sequence defined by \( a_n = \left(1 - \frac{2}{n}\right)^n \). Our goal is to find the limit as \( n \to \infty \).

Apply Limit Definition

To find the limit, let's rewrite \( a_n \) as an exponential function: \[ a_n = \exp\left( n \ln\left( 1 - \frac{2}{n} \right) \right) \] Using the properties of logarithms.

Use Logarithmic Approximation

For small \( x \), \( \ln(1 - x) \approx -x \). Thus, \[ \ln\left( 1 - \frac{2}{n} \right) \approx -\frac{2}{n} \] Therefore, our expression for \( a_n \) becomes: \[ a_n = \exp\left( n \cdot \left(-\frac{2}{n}\right) \right) = \exp(-2) \]

Conclude the Limit

As \( n \to \infty \), the expression simplifies to: \[ \lim_{{n \to \infty}} a_n = \exp(-2) = \frac{1}{e^2} \] This indicates that the sequence converges.

Key concepts

Sequence Limits

When dealing with sequences, understanding their limits is crucial. A sequence is simply an ordered list of numbers, and it converges if it approaches a specific value as the number of terms increases. In the exercise given, we looked at the sequence defined by \( a_n = \left(1 - \frac{2}{n}\right)^n \). The limit of this sequence as \( n \to \infty \) tells us the value it approaches.

Identifying the limit involves a deeper inspection of the sequence's behavior for very large values of \( n \). If there is a single value that the sequence consistently approaches, then it converges. If not, it diverges. A convergent sequence has a limit, while a divergent sequence does not. In our exercise example, the sequence converges to \( \frac{1}{e^2} \), indicating a specific predictable behavior as \( n \to \infty \).

Understanding these limits is essential because they help us predict long-term behaviors in mathematical models, whether in economics, physics, or any field that uses mathematical modeling.

Logarithmic Approximation

Logarithmic approximation is a valuable mathematical tool, especially useful when evaluating expressions involving logarithms for values close to one. In the problem, this technique was vital in simplifying the expression \( \ln\left( 1 - \frac{2}{n} \right) \).

For very small values of \( x \), such as in our exercise where \( x = \frac{2}{n} \) becomes small when \( n \) is large, the logarithmic approximation \( \ln(1 - x) \approx -x \) becomes applicable. This is because the function \( \ln(1 - x) \) is approximately linear for small \( x \), making \( -x \) an excellent estimate for its value.

This approximation simplifies calculations and provides a faster way to solve complex problems. Using this method allows us to determine the behavior of the sequence as \( n \) grows larger without cumbersome computations. In our specific case, it helped transform the sequence into a form that's easier to evaluate, ultimately leading to the conclusion that \( a_n \,\to\, \exp(-2) \) as \( n \to \infty \).

• Useful for simplifying expressions.
• Applicable for values close to one.
• Makes computation more efficient.

Exponential Functions

Exponential functions form a foundational concept in mathematics, characterized by a constant base raised to a variable exponent. They appear in various fields due to their ability to model growth and decay processes efficiently.

In the given exercise, we consider the exponential function expressed as \( a_n = \exp\left( n \ln\left( 1 - \frac{2}{n} \right) \right) \). Rewriting the sequence in this exponential form helps us explore the limit properties more deeply.

The expression \( \exp(x) \) represents the exponential function with base \( e \), an irrational number approximately equal to 2.718. In this exercise, it simplifies our sequence, allowing us to work with the exponential properties efficiently and understand the sequence's behavior as \( n \) increases.

• Models natural growth or decay.
• Simplifies complex equations.
• Integral for evaluating limits.

Understanding these properties aids in recognizing why \( \lim_{{n \to \infty}} a_n = \exp(-2) = \frac{1}{e^2} \). This insight into exponential functions ensures we can solve similar problems and understand the broader implications of their behavior in mathematical modeling.