Problem 47 - 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. $$ a_{n}=\sqrt{n} $$

Step-by-Step Solution

Answer

The first five terms are 0, 1, √2, √3, and 2. The limit does not exist.

Step 1: Determine Sequence Terms

To find the first five terms of the sequence \(a_n = \sqrt{n}\), substitute \(n = 0, 1, 2, 3, \text{ and } 4\) into the equation. This gives us: - \(a_0 = \sqrt{0} = 0\) - \(a_1 = \sqrt{1} = 1\) - \(a_2 = \sqrt{2} \approx 1.414\) - \(a_3 = \sqrt{3} \approx 1.732\) - \(a_4 = \sqrt{4} = 2\).So, the first five terms are: 0, 1, \(\sqrt{2}\), \(\sqrt{3}\), 2.

Step 2: Analyze the Limit

Determine whether the limit exists as \(n\) approaches infinity. The term \(a_n = \sqrt{n}\) becomes unbounded because \(\sqrt{n}\) tends to infinity as \(n\) becomes very large. Therefore, the limit \(\lim_{n\to\infty} \sqrt{n}\) does not exist.

Key Concepts

Infinite Sequences

An infinite sequence is a list of numbers written in a specific order, extending indefinitely. Each number in this list is called a 'term.' For example, in the sequence defined by \(a_n = \sqrt{n}\), the terms are generated by plugging consecutive natural numbers into the formula. So, the sequence starts with 0, 1, \(\sqrt{2}\), \(\sqrt{3}\), 2, and continues indefinitely as \(n\) increases.

• Key Property: Infinite sequences do not have a final term. Instead, they keep going, allowing us to examine them as \(n\) gets larger with no upper limit.
• Application: These sequences are crucial in calculus and mathematical analysis because they set the stage for understanding functions that stretch indefinitely.
• Visualization: Picture an infinite sequence on a number line. Starting at a point, you can imagine moving along without ever stopping or reaching an endpoint.

Understanding infinite sequences helps us delve into more complex mathematical concepts such as limits and convergence.

Sequence Convergence

In mathematics, a sequence is said to "converge" if it approaches a specific value as \(n\) becomes very large. This specific value is known as the "limit" of the sequence. For a converging sequence, the terms get closer and closer to the limit. However, not all sequences have limits. For example, in the sequence \(a_n = \sqrt{n}\), we observe that as \(n\) increases, the sequence does not converge to a fixed number. Instead, it continues to grow without bounds.

• Definition: A sequence \(a_n\) converges to a limit \(L\) if, for every positive number \(\epsilon\), there exists an \(N\) such that whenever \(n > N\), \(|a_n - L| < \epsilon\).
• Example: The sequence \(a_n = 1/n\) converges to 0 as \(n\) approaches infinity.
• Key Note: The sequence \(a_n = \sqrt{n}\) is divergent—it lacks convergence because it doesn't tend to any particular number as \(n\) grows.

Analyzing sequence convergence is pivotal in calculus, forming the basis for discussing series and their sums.

Asymptotic Behavior

Asymptotic behavior describes how a function or sequence behaves as its variable approaches a certain point, often infinity. When studying sequences, asymptotic behavior provides insight into the 'end behavior' of the sequence. Consider again the sequence \(a_n = \sqrt{n}\). Here, as \(n\) increases, \(\sqrt{n}\) also increases, showing an unbounded growth. It suggests that the sequence diverges as it does not approach any finite value.

• Importance: Understanding the asymptotic behavior helps identify whether a function or sequence levels off, shoots towards infinity, or has other end behavior characteristics.
• Visual Aid: Plotting \(a_n\) against \(n\) typically shows a graph that rises steadily, highlighting its unbounded growth.
• Example: The asymptotic behavior of \(\log(n)\) grows slowly but tends towards infinity as \(n\) increases.

In calculus, studying asymptotic behavior aids in predicting and comparing the growth rates of functions.