Problem 57 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(1+\frac{7}{n}\right)^{n} $$

Step-by-Step Solution

Answer

The sequence \( a_n \) converges to \( e^7 \).

Step 1: Analyze the Sequence

We need to determine the behavior of the sequence \( a_n = \left(1 + \frac{7}{n}\right)^n \). This form suggests a potential convergence to \( e^c \) for some constant \( c \) as \( n \to \infty \).

Step 2: Identify the Limit Using Limit Properties

We recognize this sequence as resembling the definition of the exponential constant \( e \), specifically the limit \( \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x \). In this case, \( x = 7 \).

Step 3: Apply the Exponential Limit Property

Applying the limit property, we find \( \lim_{n \to \infty} \left(1 + \frac{7}{n}\right)^n = e^7 \).

Step 4: Conclusion on Convergence

Since a limit exists and is finite (i.e., \( e^7 \)), the sequence \( a_n \) converges to \( e^7 \).

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value. This type of growth is common in nature, finance, and technology. For example, populations, investments, or technological advances often exhibit exponential growth.

In mathematics, when a sequence grows exponentially, it means each term in the sequence is a constant multiple or raised to a power of the previous term. This is a faster type of growth compared to linear growth, where changes are constant. The general form of such a function is:

• For population growth: \( P(t) = P_0 e^{kt} \)
• For geometric sequences: \( a_n = a_1 r^{n-1} \)

Exponential growth in sequences plays a vital role in calculus, as it often involves limits and convergence, which are crucial for understanding how sequences behave as they extend indefinitely.

Limit of a Sequence

The concept of the limit of a sequence is fundamental in calculus. A sequence's limit is the value that its terms approach as the term number becomes infinitely large. Essentially, this means that given any number, no matter how small, we can find a term in the sequence such that all subsequent terms are arbitrarily close to the limit.

For a sequence \( \{ a_n \} \) to have a limit \( L \), it must hold true that:

• \( \forall \epsilon > 0, \exists N \in \mathbb{N} \) such that if \( n > N \), then \( |a_n - L| < \epsilon \)

This definition implies that after a certain point, all terms of the sequence are within an \( \epsilon \) distance of the limit. Taking the given exercise as an example, the sequence \( a_n = \left(1 + \frac{7}{n}\right)^n \) approaches the limit \( e^7 \) as \( n \rightarrow \infty \). Understanding limits helps in analyzing the behavior and convergence of sequences.

Convergence and Divergence

In calculus, determining whether a sequence converges or diverges is key to understanding its behavior. A sequence converges if it approaches a specific limit as the term number goes to infinity. If no such limit exists, the sequence is divergent.

For example, the sequence \( a_n = \left(1 + \frac{7}{n}\right)^n \) converges to \( e^7 \). This is because, as \( n \) becomes very large, the terms get closer to \( e^7 \). Conversely, if a sequence’s terms do not settle into specific behavior or continue to grow without bound, it is divergent.

• Convergent sequences: Reach a finite limit.
• Divergent sequences: Do not settle on a limit; they may oscillate or increase indefinitely.

Recognition of these behaviors is crucial not only in sequence analysis but also in series, integrals, and other areas of advanced calculus.

Calculus Sequences

Calculus sequences are ordered lists of numbers, following particular rules, crucial to understanding series and functions. In calculus, these sequences often portray complex behaviors as they extend toward infinity - being the foundation for many calculus concepts such as limits, continuity, and series.

The properties and limits of these sequences can be used to calculate sums of series, find derivatives, and even solve differential equations. For instance, arithmetic sequences grow by consistent differences, whereas geometric sequences grow by constant ratios. The understanding of these differences is pivotal in solving calculus problems.In our exercise, the sequence \( a_n = \left(1 + \frac{7}{n}\right)^n \) exemplifies how sequences are analyzed in calculus to determine if they converge to a limit, like \( e^7 \) in this case. By utilizing the exponential growth property and limit definitions, calculus combines with sequences to explore behaviors at the approach of infinity, pivotal for applications across mathematics and engineering.