Problem 38 - 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 \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{2}{n+1} $$

Step-by-Step Solution

Answer

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

Step 1: Determine the First Term

To find the first term, substitute \( n = 0 \) into the given formula: \( a_0 = \frac{2}{0+1} = 2 \).

Step 2: Calculate the Second Term

Substitute \( n = 1 \) into the sequence formula: \( a_1 = \frac{2}{1+1} = 1 \).

Step 3: Find the Third Term

Substitute \( n = 2 \) into the sequence formula: \( a_2 = \frac{2}{2+1} = \frac{2}{3} \).

Step 4: Calculate the Fourth Term

Substitute \( n = 3 \) into the formula: \( a_3 = \frac{2}{3+1} = \frac{1}{2} \).

Step 5: Determine the Fifth Term

For \( n = 4 \), substitute into the formula: \( a_4 = \frac{2}{4+1} = \frac{2}{5} \).

Step 6: List the First Five Terms

The first five terms of the sequence are: \( a_0 = 2 \), \( a_1 = 1 \), \( a_2 = \frac{2}{3} \), \( a_3 = \frac{1}{2} \), and \( a_4 = \frac{2}{5} \).

Step 7: Find the Limit as \(n\) Approaches Infinity

To find \( \lim_{n \to \infty} a_n \), evaluate \( \lim_{n \to \infty} \frac{2}{n+1} \). As \( n \to \infty \), the value of \( \frac{2}{n+1} \to 0 \), because the denominator becomes infinitely large compared to the constant numerator.

Key Concepts

Sequence Terms

A sequence is a list of numbers arranged in a specific order. In our exercise, we are dealing with a sequence defined by the formula \(a_n = \frac{2}{n+1}\). To understand this better, let’s explore the concept of sequence terms.
The numbers in the sequence are called terms. Each term is represented by its index \(n\) in the sequence. Here’s how you find the sequence terms:

• Start with the initial value: \(n = 0\). This gives you the first term, which is \(a_0\).
• Increment \(n\) to find subsequent terms: \(a_1, a_2, \ldots\), and so on.

For any sequence, you can find as many terms as needed by continuing this process. In our example,

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

The value of each term gives us a glimpse of how the sequence behaves.

Convergent Sequence

A sequence is considered convergent if its terms approach a specific number as \(n\) (the term position) goes to infinity. This specific number, if it exists, is known as the limit of the sequence.
To determine convergence for the sequence \(a_n = \frac{2}{n+1}\), you need to compute the limit as \(n\) approaches infinity:

• Evaluate \(\lim_{n \to \infty} \frac{2}{n+1}\).
• As \(n\) grows larger and larger, \(n+1\) also increases, making \(\frac{2}{n+1}\) smaller and moving closer to zero.

Thus, we conclude that:

• The sequence \(a_n = \frac{2}{n+1}\) converges to 0.

This convergence suggests that as we consider more and more terms in the sequence, they get arbitrarily close to 0, illustrating the idea of a convergent sequence.

Infinite Series

An infinite series involves adding up all the terms of a sequence without bound, extending indefinitely. Although the exercise primarily focuses on the sequence \(a_n = \frac{2}{n+1}\), the concept of infinite series is closely related, especially when discussing whether a series converges or diverges.
In simpler terms, if you were to consider an infinite series like \(\sum_{n=0}^{\infty} a_n\), it involves summing the terms indefinitely:

• Each term contributes a portion to the sum.
• The main question would be whether this sum remains finite or grows indefinitely.

Although calculating an infinite series requires specific techniques and does not directly apply to this problem, it is crucial to understanding much larger patterns in mathematics. It illustrates how sequences and their limits can lead into more complex and intriguing mathematical concepts.