Problem 52 - 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. $$ a_{n}=\left(\frac{1}{3}\right)^{n} $$

Step-by-Step Solution

Answer

The sequence is \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}\) and the limit is 0.

Step 1: Understand the Sequence

The given sequence is defined as \(a_n = \left(\frac{1}{3}\right)^{n}\). This means that each term is the result of raising \(\frac{1}{3}\) to the power of \(n\).

Step 2: Calculate the First Five Terms

To find the first five terms, we replace \(n\) with numbers ranging from 0 to 4. 1. When \(n = 0\), \(a_0 = \left(\frac{1}{3}\right)^0 = 1\).2. When \(n = 1\), \(a_1 = \left(\frac{1}{3}\right)^1 = \frac{1}{3}\).3. When \(n = 2\), \(a_2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9}\).4. When \(n = 3\), \(a_3 = \left(\frac{1}{3}\right)^3 = \frac{1}{27}\).5. When \(n = 4\), \(a_4 = \left(\frac{1}{3}\right)^4 = \frac{1}{81}\).

Step 3: Identify Terms

Using the calculations from the previous step, the first five terms of the sequence are: \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}\).

Step 4: Determine the Limit of the Sequence

The limit \(\lim_{n \to \infty} a_n\) can be found by examining the behavior of \(\left( \frac{1}{3} \right)^n\) as \(n\) becomes very large. Since \(\frac{1}{3}\) is a fraction between 0 and 1, its powers \(\left( \frac{1}{3} \right)^n\) approach 0 as \(n\) increases indefinitely. Thus, the limit exists and \(\lim_{n \to \infty} \left( \frac{1}{3} \right)^n = 0\).

Key Concepts

Exponential Sequence

In mathematics, an exponential sequence is a type of sequence where each term is a constant raised to a power that varies with the position in the sequence. In this exercise, the sequence is defined as \(a_n = \left(\frac{1}{3}\right)^n\). Here, \(\frac{1}{3}\) is the constant base and \(n\) is the changing exponent representing the sequence's term number.

An exponential sequence can demonstrate growth or decay depending on the base's value:

• If the base is greater than 1, the sequence grows exponentially.
• If the base is between 0 and 1, the sequence decreases exponentially, as seen in this exercise.

Recognizing the characteristics of an exponential sequence is crucial for understanding how the sequence will behave as \(n\) changes.

Limits of Functions

The limit of a sequence refers to the value the sequence approaches as the index \(n\) goes to infinity. In our sequence, we investigated the terms as \(n\) increased. Each term was of the form \(\left(\frac{1}{3}\right)^n\).

To assess the limit, note that since \(\frac{1}{3}\) is a fraction less than one, raising it to an increasingly large power results in values that get smaller and smaller, approaching zero. Therefore, the sequence \(a_n = \left(\frac{1}{3}\right)^n\) gets closer to zero. Thus, we say that the limit of the sequence as \(n\) approaches infinity is 0:

\[\lim_{n \to \infty} \left(\frac{1}{3}\right)^n = 0\] Understanding limits is key to determining the long-term behavior of sequences and functions.

Convergent Sequences

A sequence is convergent if it approaches a specific finite number as \(n\) becomes infinitely large. In simpler words, a sequence converges when the terms get arbitrarily close to a particular value.

In this context, the sequence \(a_n = \left(\frac{1}{3}\right)^n\) is convergent because, over time, its terms get closer and closer to zero, an increasingly narrower neighborhood around zero. The steady movement toward zero shows that the sequence has a limit—a defining feature of convergence.

Identifying whether a sequence converges helps in various applications where predicting behavior at infinity is essential, such as in calculus and real-world modeling. Recognizing convergence simplifies the complexity of sequences by capturing their long-term behavior in a single number.