Problem 45 - Calculus for Biology and Medicine

Question

In Problems \(45-52\), write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=\frac{n^{2}}{n+1} $$

Step-by-Step Solution

Answer

The first five terms are 0, 1/2, 4/3, 9/4, 16/5. The sequence diverges as n approaches infinity.

Step 1: Calculate the First Term

Substitute \(n = 0\) into the sequence formula: \[ a_0 = \frac{0^2}{0+1} = \frac{0}{1} = 0 \] The first term is \(a_0 = 0\).

Step 2: Calculate the Second Term

Substitute \(n = 1\) into the sequence formula: \[ a_1 = \frac{1^2}{1+1} = \frac{1}{2} \] The second term is \(a_1 = \frac{1}{2}\).

Step 3: Calculate the Third Term

Substitute \(n = 2\) into the sequence formula: \[ a_2 = \frac{2^2}{2+1} = \frac{4}{3} \] The third term is \(a_2 = \frac{4}{3}\).

Step 4: Calculate the Fourth Term

Substitute \(n = 3\) into the sequence formula: \[ a_3 = \frac{3^2}{3+1} = \frac{9}{4} \] The fourth term is \(a_3 = \frac{9}{4}\).

Step 5: Calculate the Fifth Term

Substitute \(n = 4\) into the sequence formula: \[ a_4 = \frac{4^2}{4+1} = \frac{16}{5} \] The fifth term is \(a_4 = \frac{16}{5}\).

Step 6: Determine the Behavior of the Sequence as n Approaches Infinity

To find the limit as \(n\) approaches infinity, use the expression: \[ \lim_{{n \to \infty}} \frac{n^2}{n+1} \]Divide the numerator and the denominator by \(n\):\[ \lim_{{n \to \infty}} \frac{n}{1 + \frac{1}{n}} \]As \(n\) approaches infinity, \(\frac{1}{n}\) approaches 0, so:\[ \lim_{{n \to \infty}} \frac{n}{1 + 0} = n \]The limit does not exist as the sequence approaches infinity.

Key Concepts

Sequence Terms

In the world of sequences, the term "sequence terms" refers to the individual elements that make up a sequence. A sequence is an ordered list of numbers, often defined by a specific formula. These terms help us understand the behavior and properties of the sequence.

In the given exercise, the sequence is defined by the formula:
\[ a_{n} = \frac{n^{2}}{n+1} \]
This formula tells us how to calculate each term, starting from when \( n=0 \). To find the first few terms, simply substitute different values of \( n \) into the formula:

• For \( n = 0 \), the term is \( a_0 = 0 \).
• For \( n = 1 \), the term is \( a_1 = \frac{1}{2} \).
• For \( n = 2 \), the term is \( a_2 = \frac{4}{3} \).
• For \( n = 3 \), the term is \( a_3 = \frac{9}{4} \).
• For \( n = 4 \), the term is \( a_4 = \frac{16}{5} \).

By calculating these terms, we can observe the pattern and growth of the sequence. Each term gets progressively larger as \( n \) increases.

Convergence and Divergence

Convergence and divergence are fundamental concepts in analyzing sequences. They describe how a sequence behaves as more terms are included, particularly as the sequence extends towards infinity.

• A sequence converges if its terms approach a specific finite limit. For example, if a sequence levels off to a particular number as \( n \) increases, it is said to converge.
• Conversely, a sequence diverges if its terms do not settle at any one value. This can happen if the terms grow without bound or if they oscillate infinitely without approaching a fixed number.

In the provided exercise, we are tasked with determining whether the sequence \( a_{n} = \frac{n^{2}}{n+1} \) converges or diverges. By examining the limit as \( n \) approaches infinity, we discover that:As \( n \) becomes very large, the fraction \( \frac{n^2}{n+1} \) simplifies to \( \frac{n}{1} = n \). Since \( n \) grows without bound, the sequence does not converge to a specific finite value; hence, it diverges.

Infinite Limits

An infinite limit describes the behavior of a sequence or function as it grows without bound. This is crucial for understanding sequences that do not converge to a finite number.

In the sequence \( a_{n} = \frac{n^{2}}{n+1} \), we calculated the limit as \( n \) approaches infinity. Initially, you might try dividing both the numerator and the denominator by \( n \) to simplify the expression:\[\lim_{{n \to \infty}} \frac{n^2}{n+1} =\lim_{{n \to \infty}} \frac{n}{1 + \frac{1}{n}}\]As \( n \) gets larger, \( \frac{1}{n} \) approaches zero, leading to:\[\lim_{{n \to \infty}} \frac{n}{1} = n\]This result tells us the sequence keeps increasing and does not settle at any finite value, hence it has an infinite limit.

Understanding that the sequence has an infinite limit is critical in recognizing why the sequence diverges. This knowledge can be applied to analyze other sequences and their behavior at infinity.