Problem 48 - 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. $$ a_{n}=n^{2} $$

Step-by-Step Solution

Answer

The first five terms are 0, 1, 4, 9, 16, and the limit does not exist.

Step 1: Identify the Terms of the Sequence

The given sequence is defined as \(a_{n} = n^2\). To find the first five terms, substitute the first five natural numbers for \(n\).

Step 2: Compute the First Five Terms

Calculate the value of \(a_{n}\) for \(n = 0, 1, 2, 3, 4\):- For \(n = 0\), \(a_0 = 0^2 = 0\)- For \(n = 1\), \(a_1 = 1^2 = 1\)- For \(n = 2\), \(a_2 = 2^2 = 4\)- For \(n = 3\), \(a_3 = 3^2 = 9\)- For \(n = 4\), \(a_4 = 4^2 = 16\)Thus, the first five terms of the sequence are \(0, 1, 4, 9, 16\).

Step 3: Determine the Limit of the Sequence

Consider the long-term behavior of \(a_n = n^2\) as \(n\) approaches infinity. The expression \(n^2\) grows without bound since the square of any real number increases indefinitely as \(n\) increases.

Step 4: Evaluate the Existence of the Limit

A sequence has a limit \(L\) if the terms of the sequence approach \(L\) as \(n\) goes to infinity. However, because \(n^2\) increases without bound, the sequence \(a_n = n^2\) does not approach any finite limit. Thus, \(\lim_{n \to \infty} a_n\) does not exist.

Key Concepts

Sequence Terms

In mathematics, a sequence is essentially a list of numbers written in a specific order. Each number in this list is called a "term," and the order in which the terms appear is important. The given sequence \( \{a_{n}\} \) is defined with a specific rule: \( a_n = n^2 \). This means each term is the square of its position number, starting from 0.

To break it down simply, let's understand through examples:

• For \( n = 0 \), \( a_0 = 0^2 = 0 \)
• For \( n = 1 \), \( a_1 = 1^2 = 1 \)
• For \( n = 2 \), \( a_2 = 2^2 = 4 \)
• For \( n = 3 \), \( a_3 = 3^2 = 9 \)
• For \( n = 4 \), \( a_4 = 4^2 = 16 \)

These calculated values represent the first five terms of the sequence. Each term gets progressively larger as we move through the sequence.

Infinite Sequence

An infinite sequence is a sequence that continues indefinitely. Unlike a finite sequence that stops after a certain number of terms, an infinite sequence has no last term. This concept is critical when discussing limits and analyzing the behavior of sequences as they extend towards infinity.

For the given sequence \( a_{n} = n^2 \), as \( n \) grows larger and larger, the values of \( a_n \) also increase without stopping. There isn't a final term; instead, the sequence extends infinitely. This type of sequence is essential in mathematical analysis and helps in understanding the concept of limits and convergence of series.

Understanding infinite sequences is vital, particularly when determining if they approach a finite limit or continue to grow indefinitely.

Squared Sequence

A squared sequence refers to a sequence in which each term is the square of a specific number. In our example, the sequence \( a_{n} = n^2 \) is a classic squared sequence. Each term grows according to the square of its index, \( n \). This characteristic is key in understanding the behavior of the sequence.

What does squaring mean in this context? When each index \( n \) is squared, it causes the values to increase rapidly compared to linear growth. For example:

• At \( n = 1 \), \( a_1 = 1 \)
• At \( n = 2 \, \) \( a_2 = 4 \)
• At \( n = 3 \), \( a_3 = 9 \)
• At \( n = 4 \), \( a_4 = 16 \)

The rapid increase in values explains why the sequence doesn't converge to a limit, consistently increasing with each subsequent term. This property shows how powerful the function \( n^2 \) is in driving the sequence to grow indefinitely.