Problem 47 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = \left( 1+ \frac {2}{n} \right)^n \)

Step-by-Step Solution

Answer

The sequence converges to \( e^2 \).

Step 1: Identify the Type of Sequence

This sequence, given by \( a_n = \left( 1+ \frac{2}{n} \right)^n \), is an exponential sequence that resembles the form of sequences that converge to the exponential constant \( e \). Due to its structure, it's helpful to compare it to the general form \( \left( 1 + \frac{1}{n} \right)^n \), which converges to \( e \).

Step 2: Apply the Exponential Limit Property

The sequence \( a_n \) is similar to the expression \( \left( 1 + \frac{k}{n} \right)^n \), which converges to \( e^k \) as \( n \to \infty \). In our sequence, \( k = 2 \). Therefore, as \( n \to \infty \), \( \left( 1 + \frac{2}{n} \right)^n \to e^2 \).

Step 3: Demonstrate Convergence Using Logarithms

Consider \( \ln(a_n) = n \cdot \ln \left( 1 + \frac{2}{n} \right) \). Using the approximation \( \ln(1+x) \approx x \) for small \( x \), we can write \( \ln(a_n) \approx n \cdot \frac{2}{n} = 2 \). Thus, \( a_n = e^{\ln(a_n)} \approx e^2 \).

Step 4: Conclusion on Convergence and Limit

Given the analysis and the application of the exponential limit property and logarithmic approximation, we conclude the sequence \( a_n = \left( 1+ \frac{2}{n} \right)^n \) converges to \( e^2 \) as \( n \to \infty \).

Key Concepts

Exponential Sequences

Exponential sequences are sequences where each term is raised to the power corresponding to its index or follows a pattern involving exponentiation. The sequence in the exercise, given by \( a_n = \left( 1+ \frac{2}{n} \right)^n \), is a classic example of an exponential sequence. In such sequences, the value of each term depends not only on a base number but often involves terms from an index formula, such as \( 1 + \frac{k}{n} \).

These sequences can be fascinating because even minor changes in the formula, like the term \( \frac{2}{n} \), significantly affect the convergence and behavior of the sequence. Understanding the basic setup of exponential sequences can help you predict their limits and behavior as they progress towards infinity. With practice, identifying these patterns gets easier, and you can apply your knowledge to similar situations involving mathematical growth or decay.

• Base sequence pattern: \( \left( 1+ \frac{1}{n} \right)^n \)
• Modification pattern: \( 1 + \frac{k}{n} \) where \( k \) is a constant

Exponential Limit Property

The exponential limit property is the principle describing the tendency of certain exponential sequences to converge as their index approaches infinity. This is particularly relevant for sequences of the form \( \left( 1 + \frac{k}{n} \right)^n \), which approach an exponential function \( e^k \) when \( n \longrightarrow \infty \).

To apply this property to our sequence, \( a_n = \left( 1+ \frac{2}{n} \right)^n \), observe that it closely resembles the exponential form. Here, \( k = 2 \), meaning as \( n \) reaches ever larger values, the sequence converges to \( e^2 \).

• This property offers a powerful tool for determining limits of complex sequences.
• It links sequence behavior to known constants like \( e \), simplifying the analysis of sequence convergence.

Logarithmic Approximation

Logarithmic approximation is a useful technique in mathematics for simplifying complex expressions, especially when dealing with limits. When considering the sequence \( a_n = \left( 1+ \frac{2}{n} \right)^n \), you can use logarithmic approximation to analyze its convergence. The natural logarithm \( \ln(1+x) \) can be approximated as \( x \) when \( x \) is small, which is particularly useful here.

We calculate the natural logarithm of the sequence's terms: \( \ln(a_n) = n \cdot \ln \left( 1 + \frac{2}{n} \right) \). Using the approximation \( \ln(1+x) \approx x \), substitute to get \( \ln(a_n) \approx n \cdot \frac{2}{n} = 2 \). Consequently, this simplifies the computation, showing that \( a_n \approx e^2 \).

• Approximations make the arithmetic more tractable, especially for limits.
• Offers a practical route to understanding convergence without cumbersome calculations.

Limit of a Sequence

The limit of a sequence refers to the value that the terms of a sequence approach as the index becomes very large. In essence, if a sequence \( a_n \) converges, the terms get arbitrarily close to a specific number as \( n \) increases. In this exercise, we determined the limit of the sequence \( a_n = \left( 1+ \frac{2}{n} \right)^n \) was \( e^2 \).

Understanding limits is crucial for analyzing sequence behavior in calculus and advanced mathematics. It allows you to predict long-term trends of functions and sequences and is a foundational concept for further studies in mathematical analysis and continuity.

• The sequence's terms become indistinguishable from the limit beyond a certain point.
• Limits help in forming a mental model of sequence progression and eventual outcomes.