Problem 50 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{\sin ^{2} n}{2^{n}} $$

Step-by-Step Solution

Answer

The sequence \( \{a_n\} \) converges, and the limit is 0.

Step 1: Define the Sequence

The sequence given is \( a_n = \frac{\sin^2 n}{2^n} \), where \( n \) is a natural number. We want to determine if this sequence converges or diverges as \( n \to \infty \).

Step 2: Analyze the Behavior of the Numerator

The term \( \sin^2 n \) can take any value between 0 and 1 inclusively. Therefore, \( 0 \leq \sin^2 n \leq 1 \). This tells us the maximum value of the numerator is 1.

Step 3: Analyze the Behavior of the Denominator

The denominator is \( 2^n \). As \( n \to \infty \), \( 2^n \) grows very rapidly towards infinity. This rapid growth will dominate the behavior of the sequence.

Step 4: Apply the Limit

Given that \( 0 \leq \sin^2 n \leq 1 \), it follows that \( 0 \leq \frac{\sin^2 n}{2^n} \leq \frac{1}{2^n} \). As \( n \to \infty \), the sequence \( \frac{1}{2^n} \to 0 \) because \( 2^n \to \infty \). By the Squeeze Theorem, since \( \frac{\sin^2 n}{2^n} \) is squeezed between 0 and something that tends to 0, it must also tend to 0.

Step 5: Conclusion about Convergence

Since \( \frac{\sin^2 n}{2^n} \to 0 \) as \( n \to \infty \), the sequence converges. The limit of the sequence is 0.

Key Concepts

Limits of sequences

In mathematics, one of the fundamental concepts is the limit of a sequence. A sequence is a list of numbers in a specific order, and the limit is essentially what the sequence approaches as the index number, often represented as \( n \), grows larger and larger.
For example, when we say a sequence \( \{a_n\} \) converges to a limit, \( L \), it means that as \( n \to \infty \), the terms of the sequence get closer and closer to \( L \). Limiting behavior helps us understand the long-term tendencies of sequences, such as whether they stabilize to a particular value or diverge to infinity or negative infinity.
Establishing the limit of a sequence requires examining the terms \( a_n \) closely. If we can demonstrate through algebraic manipulation or leveraging other mathematical tools that \( a_n \) approaches a particular value, then that value is considered the limit of the sequence. If, however, \( a_n \) does not approach any finite value, the sequence is considered divergent and does not have a limit.

Squeeze Theorem

The Squeeze Theorem is an essential tool in calculus that helps establish the limit of a sequence when it is particularly challenging to compute directly. This theorem essentially states that if a sequence \( \{a_n\} \) is "squeezed" between two other sequences \( \{b_n\} \) and \( \{c_n\} \), and both \( \{b_n\} \) and \( \{c_n\} \) converge to the same limit, then \( \{a_n\} \) must also converge to that same limit.
In practical terms, if you have a sequence \( a_n \) such that \( b_n \leq a_n \leq c_n \) for all \( n \), and \( \lim_{{n\to\infty}} b_n = \lim_{{n\to\infty}} c_n = L \), then \( \lim_{{n\to\infty}} a_n = L \) as well. This rule is especially useful when dealing with sequences that involve terms like sine functions combined with exponential growth, as we saw in the example problem.

• It provides a method of simplifying our work by allowing us to focus on approximating and bounding terms rather than finding complex direct limits.
• This theorem is widely applicable—whenever direct computation of a limit seems difficult, the Squeeze Theorem might just save the day.

Exponential functions

Exponential functions are a key topic in mathematics, defined broadly as functions of the form \( f(x) = a^x \), where \( a \) is a constant. They exhibit very rapid growth or decay depending on the base \( a \):

• If \( a > 1 \), the function describes exponential growth, meaning it increases at an ever-accelerating rate as the exponent rises.
• If \( 0 < a < 1 \), the function models exponential decay, where it rapidly decreases towards zero as the exponent increases.

In the context of sequences and convergence, exponential functions often affect the denominator of a fraction, as in the given sequence \( a_n = \frac{\sin^2 n}{2^n} \), where the sequence goes to zero because the exponential denominator grows far faster than the numerator.
The growth rate of exponential functions makes them powerful tools in modeling scenarios ranging from population growth to radioactive decay, offering insights into how certain systems evolve over time. Understanding exponential behavior is crucial for managing and predicting dynamic changes in real-world problems. Yet, this very property makes them useful in solving convergence issues by simply explaining why certain numerators fade into insignificance against substantial denominators like \( 2^n \).