Problem 56 - 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 determine whether \(\lim _{n \rightarrow \infty} a_{n}\) exists. If the limit exists, find it. $$ a_{n}=n^{2}+3 $$

Step-by-Step Solution

Answer

The first five terms are 3, 4, 7, 12, and 19. The limit does not exist.

Step 1: Write General Term

The formula for the sequence given is \( a_n = n^2 + 3 \). This formula allows us to calculate the terms of the sequence by substituting different values of \( n \).

Step 2: Calculate First Term

To find the first term of the sequence, set \( n = 0 \). Therefore, \( a_0 = 0^2 + 3 = 3 \).

Step 3: Calculate Second Term

Proceed to find the second term by setting \( n = 1 \). This gives \( a_1 = 1^2 + 3 = 4 \).

Step 4: Calculate Third Term

Set \( n = 2 \) to find the third term. Thus, \( a_2 = 2^2 + 3 = 7 \).

Step 5: Calculate Fourth Term

For the fourth term, set \( n = 3 \). We have \( a_3 = 3^2 + 3 = 12 \).

Step 6: Calculate Fifth Term

Finally, calculate the fifth term by setting \( n = 4 \). Therefore, \( a_4 = 4^2 + 3 = 19 \).

Step 7: Analyze the Limit

Consider the expression \( a_n = n^2 + 3 \) as \( n \) approaches infinity. As \( n \) becomes very large, \( n^2 \) dominates the constant \( 3 \), causing \( a_n \to \infty \). This means the limit \( \lim_{n \to \infty} a_n \) does not exist.

Key Concepts

Infinite Sequences

Infinite sequences are comparisons of ordered lists of numbers that never end. You typically see them indexed by natural numbers starting from zero, represented as \( \{a_0, a_1, a_2, a_3, a_4, \ldots \} \). The sequence in this exercise follows the formula \( a_n = n^2 + 3 \). By understanding this, you can generate any term in the sequence for specific integer values of \( n \).

For example, when calculating a few initial terms:

• Step 1: For \( n = 0 \), \( a_0 = 0^2 + 3 = 3 \)
• Step 2: For \( n = 1 \), \( a_1 = 1^2 + 3 = 4 \)
• Step 3: For \( n = 2 \), \( a_2 = 2^2 + 3 = 7 \)
• Step 4: For \( n = 3 \), \( a_3 = 3^2 + 3 = 12 \)
• Step 5: For \( n = 4 \), \( a_4 = 4^2 + 3 = 19 \)

The concept of infinite sequences is crucial as it allows mathematicians and students to understand patterns and behaviors of sequences beyond finite limitations.

Convergence and Divergence

Convergence and divergence refer to the behavior of sequences or series as they approach infinity. When we say a sequence converges, it means its terms get closer and closer to a specific number as \( n \) gets very large. On the other hand, a sequence diverges if it doesn't settle down to a single number.

In the given sequence \( a_n = n^2 + 3 \), as \( n \) increases, \( n^2 \) grows much faster than the constant 3, thus leading \( a_n \) towards infinity. Since the terms do not approach any finite number, the sequence diverges.

This understanding of convergence and divergence is essential for distinguishing whether any pattern or behavior in sequences can sum up to a finite limit or grow indefinitely.

Algebraic Sequences

Algebraic sequences are sequences defined by algebraic functions. In simple terms, each term in the sequence can be generated from a formula involving algebraic operations like addition, subtraction, multiplication, and division.

In this exercise, we work with the algebraic sequence defined by \( a_n = n^2 + 3 \). The formula consists of the algebraic operation of squaring the index \( n \) and then adding 3. This specific function shows us how the sequence behaves as \( n \) increases.

Using the algebraic formula to generate the sequence:

• Start with an index, for example, \( n = 0 \). The term would be \( a_0 = 0^2 + 3 = 3 \).
• Next, for \( n = 1 \), \( a_1 = 1^2 + 3 = 4 \).

Algebraic sequences provide a structured method to describe and predict the behavior of sequences across any values in their domain.