Problem 37 - Calculus for Biology and Medicine

Question

In Problems \(37-44\), write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{1}{n+2} $$

Step-by-Step Solution

Answer

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

Step 1: Substitute n = 0

To find the first term of the sequence, substitute \(n = 0\) into the formula for \(a_n\):\[a_0 = \frac{1}{0+2} = \frac{1}{2}\]

Step 2: Substitute n = 1

Next, substitute \(n = 1\) to find the second term:\[a_1 = \frac{1}{1+2} = \frac{1}{3}\]

Step 3: Substitute n = 2

Substitute \(n = 2\) into the formula to obtain the third term:\[a_2 = \frac{1}{2+2} = \frac{1}{4}\]

Step 4: Substitute n = 3

Now, find the fourth term by substituting \(n = 3\):\[a_3 = \frac{1}{3+2} = \frac{1}{5}\]

Step 5: Substitute n = 4

Finally, find the fifth term by substituting \(n = 4\):\[a_4 = \frac{1}{4+2} = \frac{1}{6}\]

Step 6: Identify the pattern in sequence

The first five terms of the sequence are \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}\). The pattern indicates that each term is of the form \(\frac{1}{n+2}\).

Step 7: Find the limit as n approaches infinity

As \(n\) approaches infinity, the term \(\frac{1}{n+2}\) approaches zero because the denominator becomes very large:\[\lim_{n \rightarrow \infty} \frac{1}{n+2} = 0\]

Key Concepts

Limit of a Sequence

To understand the limit of a sequence, let's consider what happens as we keep adding terms to the sequence. In simple terms, the limit of a sequence refers to the value that the sequence's terms get closer to as you continue to add more and more terms. In mathematical notation, if a sequence \( \{a_n\} \) approaches a limit \( L \), then we write it as \( \lim_{n \rightarrow \infty} a_n = L \).
In our given sequence \( a_n = \frac{1}{n+2} \), as \( n \) increases, the denominator \( n+2 \) becomes larger and larger, pushing the value of \( a_n \) closer to 0. Thus, we conclude that the limit of this sequence is 0. This concept is central in determining whether a sequence converges and how sequences behave as they progress indefinitely.

Infinite Sequences

When we talk about infinite sequences, we're discussing sequences that continue indefinitely. There isn't a final term in these sequences. Instead, they have an infinite number of terms. Such sequences are typically described by a formula that allows you to plug in any positive integer \( n \) to find specific terms.
The sequence \( a_n = \frac{1}{n+2} \) is a perfect example of an infinite sequence. We can continue to calculate terms for \( n = 5, 6, 7, \ldots \), each of which will provide a value, showing us the behavior of the sequence as it progresses further.

• Each term is part of an ongoing sequence with no end point.
• Infinite sequences allow us to discuss concepts like limits and convergence.
• Even though the sequence doesn't stop, its terms can lead to meaningful analysis like finding a limit.

Understanding infinite sequences lays the groundwork for more complex topics like series.

Convergence of Sequences

Convergence is a crucial concept in sequences, as it describes a sequence whose terms approach a specific value as \( n \) becomes extremely large. When a sequence converges, all its terms get closer and closer to a certain number, known as the limit, which we discussed earlier.
For example, in the sequence \( a_n = \frac{1}{n+2} \), we've established that \( lim_{n \to \infty} a_n = 0 \). This implies the sequence converges to 0.

• A converging sequence has a limit, and the terms progressively get near to this limit.
• Not all sequences converge; those that don't are called divergent.
• Analyzing the convergence of sequences helps in understanding series and the behaviors of complex functions.

Understanding convergence provides insight into how the sequence can be applied in practical scenarios, especially in calculus and real analysis.