Problem 91 - Algebra and Trigonometry

Question

What appears to be happening to the terms of each sequence as n gets larger? $$ a_{n}=\frac{2 n^{2}+5 n-7}{n^{3}} \quad n:[0,10,1] \text { by } a_{n}:[0,2,0.2] $$

Step-by-Step Solution

Answer

As \(n\) gets larger, the terms of the sequence \(a_{n}\) are getting smaller and are converging to 0.

Step 1: Expression Simplification

Simplify the sequence expression by dividing each term in the numerator by \(n^{3}\). This results in the expression \(\frac{2 n^{2}}{n^{3}} + \frac{5n}{n^{3}} - \frac{7}{n^{3}}\). By canceling out the common factors, we can simplify the expression further to \(2 \frac{1}{n} + 5 \frac{1}{n^{2}} - 7 \frac{1}{n^{3}}\).

Step 2: Determine Limit as n Approaches Infinity

For a sequence \(\frac{1}{n^{p}}\), where \(p > 0\), as \(n\) approaches infinity, the sequence approaches 0. So, taking the limit as \(n\) approaches infinity for our simplified expression, the terms \(2 \frac{1}{n}\), \(5 \frac{1}{n^{2}}\), and \(-7 \frac{1}{n^{3}}\) all approach 0.

Step 3: Final Statement about Sequence Behavior

After determining the limit of each term in the sequence as \(n\) approaches infinity, we find that each term is trending towards 0. In other words, as \(n\) gets larger, the terms of the sequence are getting smaller and approaching 0.

Key Concepts

Sequence Simplification

When dealing with limits of sequences, the first step is often sequence simplification. This simplifies evaluating their behavior as they grow larger. Consider a sequence like \( a_{n} = \frac{2n^2 + 5n - 7}{n^3} \). A useful technique is to reduce the complexity of this fraction by dividing each term in the numerator by \( n^3 \), which is the highest degree term in the denominator. This simplifies the sequence to a form where you can clearly see the effects of the powers of \( n \) as they grow large.

• First term: \( \frac{2n^2}{n^3} = 2 \frac{1}{n} \)
• Second term: \( \frac{5n}{n^3} = 5 \frac{1}{n^2} \)
• Third term: \( \frac{-7}{n^3} = -7 \frac{1}{n^3} \)

The simplified expression \( 2 \frac{1}{n} + 5 \frac{1}{n^2} - 7 \frac{1}{n^3} \) allows you to see the diminishing influence of each term as \( n \) increases. This process makes it easier to analyze the sequence's behavior without clutter.

Asymptotic Behavior

Asymptotic behavior refers to how a sequence behaves as its index \( n \) becomes very large. For the sequence in question, each term \( 2 \frac{1}{n}, 5 \frac{1}{n^2}, \text{and} -7 \frac{1}{n^3} \) is influenced by the denominator's growth due to increasing \( n \). Each

• \( \frac{1}{n} \) terms become very small as \( n \) grows since dividing by a larger figure results in a smaller fraction.
• The higher the exponent in the denominator, the faster the term shrinks.

Thus, as \( n \rightarrow \infty \), every component term tends towards zero. By evaluating these individually based on their powers of \( n \), you gain insight into the sequence's asymptotic trend—suggesting in this case a movement towards zero.

Infinity Limit

The concept of an infinity limit is crucial when determining the eventual behavior of a sequence. In the context of this sequence \( a_{n} = 2 \frac{1}{n} + 5 \frac{1}{n^2} - 7 \frac{1}{n^3} \), as \( n \) approaches infinity, each term individually approaches zero. This is because, fundamentally, the infinity limit of \( \frac{1}{n^p} \) where \( p > 0 \) is zero. Understanding the infinity limit allows you to see that the larger \( n \) gets:

• The term \( 2 \frac{1}{n} \) tends to get smaller and closer to zero.
• \( 5 \frac{1}{n^2} \), which shrinks even faster, also approaches zero.
• \( -7 \frac{1}{n^3} \), with its even greater exponent in the denominator, tends towards zero at an even faster rate.

By calculating the limit of each component to determine that it zeroes out, you conclude the sequence itself approaches a definitive limit of zero as \( n \) grows indefinitely large. This understanding eliminates guesswork in predicting the sequence's end behavior.