Problem 12 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether or not the sum is geometric. Assume \("+\cdots\) indicates that the established pattern continues. If the sum is geometric, identify " \(a\) " and " \(r\) ". $$ \frac{1}{e}+\frac{2}{e^{2}}+\frac{4}{e^{3}}+\cdots+\frac{2^{n}}{e^{n+1}} $$

Step-by-Step Solution

Answer

Yes, the sum \( \frac{1}{e}+\frac{2}{e^2}+\frac{4}{e^3}+\cdots+\frac{2^n}{e^{n+1}} \) is a geometric progression with 'a' = \( \frac{1}{e} \) and 'r' = 2.

Step 1: Identify the given series

Firstly, observe the given sum: \( \frac{1}{e}+\frac{2}{e^2}+\frac{4}{e^3}+\cdots+\frac{2^n}{e^{n+1}}\). You can notice that the numerator is doubling while the denominator has an extra \(e\) for each term or in other words, it is increasing its power by one for each term.

Step 2: Test geometric sequence

A geometric sequence or progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio. Therefore, we need to verify if the ratio between each pair of consecutive terms remains constant. As we observe, we get: \( \frac{2/e^2}{1/e} = 2, \frac{4/e^3}{2/e^2} = 2 \). We can see that the ratio remains constant (2), therefore, the series is geometric.

Step 3: Identify 'a' and 'r'

In a geometric progression, 'a' represents the first term of the series, and 'r' represents the common ratio. In this scenario, the first term of series 'a' is \( \frac{1}{e} \) and the common ratio 'r' is 2.

Key Concepts

Geometric Progression

A geometric progression, also known as a geometric series, is a sequence of numbers where each term after the first can be found by multiplying the previous term by a constant value known as the common ratio. To illustrate, let's take a look at an example: if we start with 3 and use a common ratio of 2, the series becomes 3, 6, 12, 24, and so on.
In the given exercise, we are presented with a series that appears to increase in a patterned manner and are asked to determine if it's geometric. Upon closer observation, the series actually has terms of the form \( \frac{2^n}{e^{n+1}} \). By applying the definition of geometric progressions to this series, we can establish its geometric nature, as the terms multiply consistently by a common ratio, which is a key characteristic of a geometric sequence.
Understanding geometric progressions is vital for various mathematical applications, from simple interest problems to more complex calculus and series convergence questions. It can also be essential in real-life situations like calculating loan payments or understanding patterns in biology.

Common Ratio

The common ratio in a geometric progression is the factor by which each term of the sequence is multiplied to get the next term. It is very important because it determines the behavior of the series—whether it converges, diverges, or oscillates.
In our exercise, we calculate the common ratio by dividing a term in the progression by its preceding term, which reveals the ratio of 2: \( \frac{2/e^2}{1/e} = 2 \). Maintaining a constant common ratio across all terms confirms the sequence's geometric nature.
The common ratio also plays a pivotal role in the formula for the sum of a geometric series and can determine the exponential growth or decay within the context of the sequence. For example, a ratio greater than 1 indicates growth, while a ratio between 0 and 1 implies decay, relevant in fields like population dynamics, economics, and physics.

Exponential Function

Exponential functions are mathematical expressions that describe phenomena whose growth rate is proportional to its current value, which can be seen in the context of our geometric progression example. In general, an exponential function is defined by the formula \( y = ab^x \), where \( a \) is a constant, \( b \) is the base representing the common ratio, and \( x \) often represents time.
In the step by step solution, we are dealing with a base of \( e \) raised to an increasing power, combined with another exponential expression in the numerator. Exponential functions are omnipresent in the natural and social sciences to represent growth or decay processes such as population growth, radioactive decay, and interest compounding. This demonstrates the interconnectedness of geometric progressions with exponential functions and highlights the importance of understanding the underlying properties of such sequences to comprehend the broader scope of exponential growth and decay.