Question

Determine the limit of the sequence or show that the sequence diverges. If it converges, find its limit. $$ a_{n}=\frac{2^{n}+3^{n}}{4^{n}} $$

Short answer

The sequence converges to 0.

Step-by-step solution

Analyze Dominant Term

In the sequence \( a_n = \frac{2^n + 3^n}{4^n} \), we observe two exponential terms: \( 2^n \) and \( 3^n \). Notice that \( 3^n \) grows faster than \( 2^n \) as \( n \) increases. Hence, \( 3^n \) is the dominant term in the numerator.

Simplify the Sequence

Given \( a_n = \frac{2^n + 3^n}{4^n} \), factor \( 3^n \) from the numerator to get: \[ a_n = \frac{3^n (\frac{2^n}{3^n} + 1)}{4^n} = \left( \frac{3}{4} \right)^n \left( \left(\frac{2}{3}\right)^n + 1 \right). \] This expression separates the dominant term \( \left(\frac{3}{4}\right)^n \) and the approaching zero term \( \left(\frac{2}{3}\right)^n \).

Determine Limit of Each Component

As \( n \to \infty \), evaluate the limits of each component: 1. \( \left(\frac{2}{3}\right)^n \to 0 \) because \( \frac{2}{3} < 1 \). 2. \( \left(\frac{3}{4}\right)^n \to 0 \) because \( \frac{3}{4} < 1 \).

Calculate Overall Limit

The sequence is \[ a_n = \left( \frac{3}{4} \right)^n \left( \left( \frac{2}{3} \right)^n + 1 \right). \] Since both \( \left( \frac{3}{4} \right)^n \rightarrow 0 \) and \( \left( \frac{2}{3} \right)^n + 1 \rightarrow 1 \) as \( n \to \infty \), the overall limit is 0. Multiplying a term going to zero with a constant approaches zero.

Key concepts

Convergence

Convergence is a concept that describes the behavior of a sequence as its terms progress towards a specific number. In mathematics, a sequence converges if its terms approach a finite number as the index becomes infinitely large. This finale is called the limit of the sequence. Determining if a sequence converges involves observing the behavior of its terms as the index grows. If the sequence settles at a particular value, it implies convergence.

For the given sequence \( a_n = \frac{2^n + 3^n}{4^n} \), we apply this concept by analyzing each component of the terms. As explained, both \( \left( \frac{2}{3} \right)^n \) and \( \left( \frac{3}{4} \right)^n \) approach zero as \( n \to \infty \), since they involve bases smaller than 1 raised to large powers. This behavior signifies that our sequence continues to diminish, ultimately converging to a limit of zero.

Understanding convergence is crucial in many areas of mathematics and helps in ascertaining stability and predictability in different systems and problems.

Exponential growth

Exponential growth refers to the rapid increase of quantities whose rate of growth is proportional to their current size. It is characterized by bases greater than 1 that multiply themselves as the exponent grows. However, in our sequence like \( a_n = \frac{2^n + 3^n}{4^n} \), there's an interesting twist.

We identify \( 3^n \) in the numerator. Initially, it seems like it might lead to exponential growth since the base is greater than 1. Yet, the whole term is part of a quotient where \( 4^n \) in the denominator also grows exponentially. The key idea here is the balance between the growing terms in the numerator and the exponentially larger denominator, \( 4^n \).

Despite potential rapid increases, division by a term like \( 4^n \), which grows faster than the numerator, results in the collapse of the overall value, moving it towards a decrease or zero, as evident in the discussed sequence. This feature counters typical exponential growth's explosive end behavior.

Dominant term analysis

Dominant term analysis is a technique used in calculus to simplify sequences or functions by focusing on the term which has the most significant impact as the variable grows. It's especially useful when dealing with polynomials or combinations of exponential terms.

For the sequence \( a_n = \frac{2^n + 3^n}{4^n} \), we utilized dominant term analysis to identify \( 3^n \) as the dominant term in the numerator. This insight was crucial since as \( n \) increases, \( 3^n \) outpaces \( 2^n \). Recognizing the dominant term allows us to approximate the sequence behavior more straightforwardly.

By factoring out \( 3^n \), we simplify the sequence to \( \left( \frac{3}{4} \right)^n \left( \left( \frac{2}{3} \right)^n + 1 \right) \), making it easier to analyze its convergence behavior. Such analysis is a powerful tool in mathematics, helping predict the outcome of complex expressions by focusing on the most influential components.