Problem 40 - Calculus for Biology and Medicine

Question

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{2 n}{n+2} $$

Step-by-Step Solution

Answer

The first five terms are 0, \(\frac{2}{3}\), 1, \(\frac{6}{5}\), \(\frac{4}{3}\). The limit is 2.

Step 1: Calculate First Term

The first term is given by \(n=0\). Substitute \(n=0\) into the expression for \(a_{n}\): \[a_{0} = \frac{2 \times 0}{0+2} = 0.\]

Step 2: Calculate Second Term

Now, we find the second term with \(n=1\). Substitute \(n=1\) into the sequence expression: \[a_{1} = \frac{2 \times 1}{1+2} = \frac{2}{3}.\]

Step 3: Calculate Third Term

Substitute \(n=2\) into the sequence expression to find the third term: \[a_{2} = \frac{2 \times 2}{2+2} = 1.\]

Step 4: Calculate Fourth Term

Calculate the fourth term by substituting \(n=3\): \[a_{3} = \frac{2 \times 3}{3+2} = \frac{6}{5}.\]

Step 5: Calculate Fifth Term

Substitute \(n=4\) into the sequence formula to get the fifth term: \[a_{4} = \frac{2 \times 4}{4+2} = \frac{8}{6} = \frac{4}{3}.\]

Step 6: Find the Limit as \(n\) Approaches Infinity

Evaluate the limit of the sequence as \(n\) approaches infinity. Simplify the expression: \[\lim_{n \to \infty} a_{n} = \lim_{n \to \infty} \frac{2n}{n+2}.\] Divide the numerator and the denominator by \(n\): \[= \lim_{n \to \infty} \frac{2}{1+\frac{2}{n}}.\] As \(n\) approaches infinity, \(\frac{2}{n}\) approaches 0, thus the limit is \(2\).

Key Concepts

Limit

A limit is a fundamental concept in calculus and mathematical analysis. It is used to describe the behavior of a function or sequence as its input approaches a certain value. When we find the limit of a sequence as it approaches infinity, we are essentially asking what value the terms of the sequence are getting closer to as the sequence progresses.
In the context of the given sequence \( a_n = \frac{2n}{n+2} \), finding the limit as \( n \) approaches infinity helps us understand the behavior of the sequence for very large values of \( n \). By dividing both the numerator and the denominator by \( n \), we simplify the expression to \( \frac{2}{1+\frac{2}{n}} \).
As \( n \) becomes very large, the term \( \frac{2}{n} \) becomes negligible, approaching 0. Therefore, the limit of the sequence becomes \( 2 \). This tells us that the sequence terms will get closer and closer to 2 as \( n \) increases without bound.

Infinity

Infinity is not a number but rather a concept that represents something that is unbounded or limitless. In mathematics, it is often used to describe a quantity that grows without limit or a state that continues indefinitely.
In sequences, when discussing the behavior of a sequence as \( n \) approaches infinity, we are considering what happens as we move far out along the sequence's terms. In the sequence formulation \( \lim_{n \to \infty} a_n \), infinity helps us examine how the sequence behaves overall.
For our sequence, \( a_n = \frac{2n}{n+2} \), evaluating the limit as \( n \) approaches infinity helps determine a stable value that the sequence approaches, even though the actual sequence terms continue indefinitely. Understanding infinity is crucial as it allows us to conceptualize limits in the context of sequences and functions.

Terms of a Sequence

The terms of a sequence are the individual elements or numbers that form the sequence. Each term in a sequence is identified by its position \( n \), starting from \( n = 0, 1, 2, \ldots \), and can be represented using a general formula.
For the sequence \( a_n = \frac{2n}{n+2} \), we calculated the first five terms by substituting the values of \( n = 0, 1, 2, 3, \) and \( 4 \) into the formula:

• \( a_0 = 0 \)
• \( a_1 = \frac{2}{3} \)
• \( a_2 = 1 \)
• \( a_3 = \frac{6}{5} \)
• \( a_4 = \frac{4}{3} \)

Each term provides a snapshot of the sequence at a particular point, and together, these terms allow us to analyze the behavior and pattern of the sequence overall. Observing how these terms change as \( n \) increases is essential for understanding long-term trends in the sequence.