Problem 51 - 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. $$ a_{n}=3^{n} $$

Step-by-Step Solution

Answer

The first five terms are 1, 3, 9, 27, 81. The limit does not exist as \( n \rightarrow \infty \).

Step 1: Calculate the first five terms of the sequence

To calculate the terms, substitute the first few integers for \( n \) into the formula \( a_n = 3^n \).- For \( n = 0 \), \( a_0 = 3^0 = 1 \).- For \( n = 1 \), \( a_1 = 3^1 = 3 \).- For \( n = 2 \), \( a_2 = 3^2 = 9 \).- For \( n = 3 \), \( a_3 = 3^3 = 27 \).- For \( n = 4 \), \( a_4 = 3^4 = 81 \).Thus, the first five terms are: 1, 3, 9, 27, 81.

Step 2: Determine if the limit exists as \( n \rightarrow \infty \)

A limit exists if as \( n \) increases, \( a_n \) approaches a specific value. Here, \( a_n = 3^n \) is an exponential function, and its value grows indefinitely large as \( n \rightarrow \infty \). Thus, \( a_n \) diverges to \( \infty \), meaning the limit does not approach a fixed number.

Key Concepts

Limits and Continuity

In mathematics, when we talk about limits and continuity, we are observing what happens to a function or sequence as certain parameters change. Specifically, limits look at what value a sequence or function tends to approach as the input grows incredibly large. In our exercise, we examine the sequence \( a_n = 3^n \). As \( n \) becomes larger, it is important to determine if \( a_n \) approaches a finite value. This understanding of limits helps us determine how the sequence behaves in the long term.
One of the key tests for limits is whether the sequence becomes arbitrarily close to a specific number as \( n \) approaches infinity. If it does, then the limit exists and the sequence is said to be **convergent**. If instead, the sequence moves further away from any particular number, it does not have a limit, and we say that it **diverges**. In our exercise, the terms of \( a_n \) grow without bound as \( n \) increases. So, \( a_n \) does not settle at any finite value, and the limit does not exist.

Exponential Functions

Exponential functions are a powerful concept in mathematics, often used to describe growth and decay processes. These functions take the form \( a^x \), where \( a \) is a positive constant. In our sequence \( a_n = 3^n \), the exponential base is 3, and the variable \( n \) is in the exponent position.
Exponential growth is characterized by rapid increases: for every increment in \( n \), the value of the function multiplies by the base (3 in this case). That's precisely why the terms of our sequence \( 1, 3, 9, 27, 81 \) grow very quickly. Each new term is 3 times larger than the previous one. * Key Features of Exponential Functions:

• They start slow but grow very fast once \( n \) is increased.
• If the base is greater than 1 (like 3 in our example), the sequence or function increases indefinitely without bound.

This nature of exponential functions makes them distinct from linear or polynomial functions, which increase at a constant or slower rate, respectively.

Divergence

Divergence in sequences and series refers to scenarios where a sequence does not approach a particular limit. In simpler terms, as we progress through the terms of the sequence, it continues to get larger or oscillates too much to settle at one number.
With respect to the sequence \( a_n = 3^n \), divergence is observed as the values do not stabilize to approach a finite number as \( n \) increases. Instead, each term significantly exceeds its predecessor, growing exponentially. * Important Points Regarding Divergence:

• A sequence that heads towards infinity is divergent.
• Divergence does not necessarily imply a chaotic sequence, but simply that the sequence does not converge to a finite limit.

Divergence is an important concept to grasp in mathematical analysis since it helps distinguish between sequences that can be effectively pinned down to a single number in their infinite term behavior and those that cannot, like \( a_n = 3^n \) in our scenario.