Problem 58 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(1-\frac{1}{n}\right)^{n} $$

Step-by-Step Solution

Answer

The sequence \( a_n = \left(1 - \frac{1}{n}\right)^n \) converges to \( \frac{1}{e} \).

Step 1: Understand the Sequence

The sequence is given by \( a_n = \left(1 - \frac{1}{n}\right)^n \). We need to determine if this sequence converges or diverges, and if it converges, find its limit.

Step 2: Recall Exponential Limit Theorem

Recognize that the expression \( \left(1 - \frac{1}{n}\right)^n \) is a standard form that can be related to the mathematical constant \( e \). As \( n \to \infty \), it is known that \( \left(1 + \frac{x}{n}\right)^n \to e^x \). In this case, \( x = -1 \).

Step 3: Apply the Exponential Limit Theorem

By applying the theorem, we substitute into \( \left(1 - \frac{1}{n}\right)^n \), yielding \( \left(1 + \frac{-1}{n}\right)^n \). Using the exponential limit, we find that this approaches \( e^{-1} \), simplifying to \( \frac{1}{e} \).

Step 4: Conclusion

Since we have evaluated the sequence as \( n \to \infty \) and found that it approaches a specific number, \( \frac{1}{e} \), \( a_n \) converges to \( \frac{1}{e} \).

Key Concepts

Limit of a Sequence

In mathematics, the limit of a sequence is the value that the terms of the sequence get closer to as the index becomes infinitely large. If such a value exists, the sequence is said to converge to this limit. Otherwise, it diverges. When we talk about a sequence converging, we're referring to each element approaching a particular number as we proceed along the sequence. For example, in the sequence given by \( a_n = \left(1 - \frac{1}{n}\right)^n \), determining whether it converges involves analyzing its behavior as \( n \) approaches infinity. Calculating the limit of sequences often requires recognizing patterns or using theorems that simplify the problem. Such is the case for sequences involving expressions in a specific form that relates closely to known mathematical behavior, like the number \( e \). Understanding limits is fundamental to grasping more complex concepts, such as series and continuity.

Exponential Limit Theorem

The exponential limit theorem is a particularly powerful tool when dealing with sequences that have the form \( \left(1 + \frac{x}{n}\right)^n \). This expression arises frequently, and as \( n \to \infty \), it approaches \( e^x \), where \( e \) is the base of the natural logarithm, approximately 2.71828. In our problem, the sequence \( \left(1 - \frac{1}{n}\right)^n \) fits this form with \( x = -1 \). Thus, we apply the theorem, transforming the expression into \( \left(1 + \frac{-1}{n}\right)^n \). According to the theorem, as \( n \) becomes very large, this sequence converges towards \( e^{-1} \), which can also be written as \( \frac{1}{e} \). By employing this theorem, complex calculations are avoided, allowing for a swift determination of the sequence's limit. Recognizing this pattern is crucial for efficiently solving convergence problems in mathematics.

Sequence and Series

Sequences and series form the cornerstone of calculus and analysis. While a sequence is simply an ordered list of numbers, a series involves summing the terms of a sequence. The vast study area focusing on these two concepts explores their properties, including convergence, divergence, and specific types of sequences and series, such as arithmetic or geometric. In our case, the sequence \( a_n = \left(1 - \frac{1}{n}\right)^n \) focuses on understanding how individual terms behave as \( n \) grows larger. This behavior is different from a series, where we would be adding up terms, but the analysis methods have similarities. Grasping the concepts of both sequences and series equips students to tackle problems across various subjects in mathematics. Having a strong foundation also prepares you for advanced topics such as infinite series, power series, and Fourier series.