Problem 92 - Algebra and Trigonometry

Question

What appears to be happening to the terms of each sequence as n gets larger? $$ a_{n}=\frac{3 n^{4}+n-1}{5 n^{4}+2 n^{2}+1} \quad n:[0,10,1] \text { by } a_{n}:[0,1,0.1] $$

Step-by-Step Solution

Answer

The terms of the sequence are getting closer and closer to \( \frac{3}{5} \) as \( n \) gets larger.

Step 1: Identify the Dominating Terms

In the given sequence \( a_{n}=\frac{3 n^{4}+n-1}{5 n^{4}+2 n^{2}+1} \), the highest power of \( n \) in both numerator and denominator is \( n^{4} \). So, as \( n \) increases, the terms with \( n^{4} \) will dominate other terms in the same region of sequence.

Step 2: Simplifying the Sequence

Considering only the dominating terms, \( n^{4} \) in both numerator and denominator, the sequence can be simplified to \( a_{n} = \frac{3 n^{4}}{5 n^{4}} \) that simplifies further to \( a_{n} = \frac{3}{5} \)

Step 3: Understand the Behavior of the Sequence as n increases

As \( n \) increases, the value of \( a_{n} \) is getting closer to the constant \( \frac{3}{5} \), since the dominating terms are equal in degree, and thus, the sequence appears to be converging to \( \frac{3}{5} \). So, the result of the sequence as \( n \) gets larger will be \( \frac{3}{5} \).

Key Concepts

Dominating Terms

When examining a sequence like \( a_{n} = \frac{3n^4 + n - 1}{5n^4 + 2n^2 + 1} \), identifying the dominating terms is crucial for understanding how the sequence behaves as \( n \) becomes very large. Dominating terms are those which have the largest exponent of \( n \) in both the numerator and denominator. Specifically, these terms will be the most influential on the value of the sequence as \( n \) increases.In our sequence, the dominating term for both the numerator and denominator is \( n^4 \). Other terms like \( n \) or \( n^2 \) grow at a slower rate compared to \( n^4 \), so they have less impact on the value of the fraction for large \( n \). By focusing on these dominating terms, we simplify the sequence and uncover its core behavior.

Asymptotic Behavior

Asymptotic behavior describes the trend of a sequence as its index, \( n \), grows towards infinity. Simplifying a sequence to its dominating terms gives us an asymptotic approximation, which helps identify the convergence point of the sequence. It describes how the sequence behaves when \( n \) becomes very large, ignoring less significant terms.In the given sequence, we simplify \( a_{n} = \frac{3n^4}{5n^4} = \frac{3}{5} \), which indicates that as \( n \) approaches infinity, the sequence's value approximates \( \frac{3}{5} \). This approximation is the asymptotic value towards which the sequence converges. Therefore, the asymptotic behavior effectively tells us that, despite other terms in the sequence, as \( n \) becomes very large, \( a_{n} \) consistently gets closer to \( \frac{3}{5} \).

Rational Functions

Rational functions, such as the one in our sequence \( a_{n} = \frac{3n^4 + n - 1}{5n^4 + 2n^2 + 1} \), are quotients of two polynomials. Analyzing rational functions often involves understanding their asymptotic behavior. When the degrees of the polynomials in the numerator and denominator are equal, as with \( n^4 \) here, the rational function converges to a constant, which is the quotient of the leading coefficients. In this case, the leading term is \( n^4 \), and the leading coefficients are 3 and 5. Hence, for large \( n \), the function converges to \( \frac{3}{5} \).Understanding rational functions is important for determining the long-term behavior of sequences, especially in finding where they converge as \( n \) becomes extremely large. By simplifying based on dominating terms, we focus on what truly impacts the function's value in the limit.