Problem 50 - 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. $$ a_{n}=\left(\frac{1}{2}\right)^{n} $$

Step-by-Step Solution

Answer

The first five terms are 1, 1/2, 1/4, 1/8, 1/16, and the limit as \( n \to \infty \) is 0.

Step 1: Understanding the Sequence

The given sequence is defined as \( a_n = \left( \frac{1}{2} \right)^n \). This means that each term in the sequence is generated by raising \( \frac{1}{2} \) to the power of \( n \). Our task is to find the first five terms of this sequence and then determine if there is a limit as \( n \) approaches infinity.

Step 2: Calculating First Five Terms

Let's compute the first five terms by substituting \( n = 0, 1, 2, 3, 4 \) into the formula:1. \( a_0 = \left( \frac{1}{2} \right)^0 = 1 \)2. \( a_1 = \left( \frac{1}{2} \right)^1 = \frac{1}{2} \)3. \( a_2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \)4. \( a_3 = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \)5. \( a_4 = \left( \frac{1}{2} \right)^4 = \frac{1}{16} \)Thus, the first five terms are: \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \).

Step 3: Determining the Limit as n Approaches Infinity

The sequence \( a_n = \left( \frac{1}{2} \right)^n \) represents a geometric sequence with a common ratio \( \frac{1}{2} \), where \( 0 < \frac{1}{2} < 1 \). As \( n \) becomes very large, the value of \( \left( \frac{1}{2} \right)^n \) approaches 0 because multiplying a fraction less than 1 by itself repeatedly makes it smaller and smaller. Thus, \( \lim_{n \to \infty} a_n = 0 \).

Step 4: Conclusion

The first five terms of the sequence are \( 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \). The limit of the sequence as \( n \to \infty \) exists and is equal to 0.

Key Concepts

Limit of a Sequence

A sequence is essentially a list of numbers generated by a specific rule or formula, like \( a_n = \left( \frac{1}{2} \right)^n \) from our exercise. When we talk about the 'limit of a sequence', we're curious about the behavior of the sequence's terms as we move further and further down the list—it gives us a sense of the sequence's end behavior.
For the geometric sequence in the exercise, the terms are each multiplied by a fixed fraction \( \frac{1}{2} \). By examining this fact, we understand that repeated multiplication by a fraction smaller than 1 results in terms that get progressively tinier. Consequently, if we calculate the limit of \( a_n \) as \( n \) becomes very large, we observe that: \[ \lim_{n \to \infty} \left( \frac{1}{2} \right)^n = 0. \]
Understanding this concept helps us determine that the sequence terms will hover closer and closer to zero, even though they never quite reach it.

Convergence

In mathematical terms, convergence means that as the terms of a sequence progress towards infinity, they approach a specific value, known as the limit. It is crucial to understand the distinction between divergence and convergence. Divergent sequences do not settle to a particular value; rather, they either go off to infinity or oscillate.
In our exercise, the sequence \( a_n = \left( \frac{1}{2} \right)^n \) displays convergence because, as calculated, its limit is 0. The terms become very small, illustrating that they are gearing closer to zero. Therefore, we can confidently say that the sequence converges to 0.
Analyzing convergence helps in predicting the behavior of sequences and understanding the stability or outcome of constant processes represented by numbers.

Infinite Series

An infinite series is the sum of a sequence that continues endlessly. It differs from a sequence since we are adding up its terms. When we evaluate such a series, we're interested in whether it sums to a specific number, known as convergence of the series, or whether it lacks a defined sum.

1. Finite vs. Infinite: In a finite series, the sum is limited and easily computable because the sequence stops. However, an infinite series like our sequence \( \sum_{n=0}^{\infty} \left( \frac{1}{2} \right)^n \) involves potentially limitless terms.
2. Geometric Series: Our sequence forms a geometric series when summed. Fortunately, in cases where the absolute value of the common ratio is less than one (here, \( r = \frac{1}{2} \)), the series converges to a limit. For this specific series, the formula for the sum \( S \) of the geometric series is: \[ S = \frac{a}{1-r}, \]
   where \( a \) is the first term, namely 1. Therefore, \( S = \frac{1}{1-\frac{1}{2}} = 2 \).

Grasping infinite series is crucial for summing sequences and recognizing patterns in numerical lists that extend indefinitely.