Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Write the sum using summation notation. $$ 0.3+0.03+0.003+0.0003+0.00003+\cdots $$

Step-by-Step Solution

Answer

The geometric series \(0.3+0.03+0.003+0.0003+0.00003+\cdots\) can be expressed in summation notation as \[ \Sigma_{n=0}^{\infty} 3(0.1)^{n+1} \]

Step 1: Identify the First Term and Common Ratio

From the given sequence, observe that the first term, \(a\), is 0.3 and that each subsequent term is multiplied by a common ratio, \(r\), of 0.1 (since each term is one decimal place to the right of the previous term). This shows us that it is indeed a geometric series.

Step 2: Write the Summation Notation

Summation notation usually is represented by a sigma (\( \Sigma \)). For this geometric series with infinite terms, the summation can be represented as: \[ \Sigma_{n=0}^{\infty} ar^n \]Replace \(a\) and \(r\) with the identified values, which leads to: \[ \Sigma_{n=0}^{\infty} 0.3(0.1)^n \]

Step 3: Simplify the Summation Notation

Finally, simplify it by writing 0.3 as \(3 \times 0.1\): \[ \Sigma_{n=0}^{\infty} 3(0.1)^{n+1} \]Here, the index starts from \(n=0\), and the term has been simplified, with 3 as the coefficient and \(0.1^{n+1}\) as the nth term.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. In this type of series:

• The first term is denoted by \(a\).
• The common ratio is denoted by \(r\).

For example, in the series provided in the exercise, 0.3, 0.03, 0.003, and so on, is a geometric series. Each term is derived by multiplying the previous term by 0.1. Geometric series have the general form \(a, ar, ar^2, ar^3, \ldots\).
In summation notation, this is often written as \(\Sigma_{n=0}^{\infty} ar^n\) where \(n\) is the term number starting at 0. This allows mathematicians and students to compactly express infinite sequences. Studying geometric series is important because they appear in various fields such as finance, physics, and computer science.

Infinite Series

An infinite series is a type of series where the sum extends indefinitely. It does not terminate, unlike a finite series that has a specific number of terms. In the case of the geometric series in the exercise, the sequence 0.3, 0.03, 0.003, and so forth continues infinitely.
An infinite series is critical in mathematics as it helps to understand behaviors closer to real-world situations where processes can continue indefinitely. Some infinite series converge, meaning they approach a specific value. Others diverge, meaning they increase indefinitely without approaching any particular value.
The concept of convergence is particularly interesting for infinite geometric series, as some can sum to a finite limit if the absolute value of the common ratio, \(|r|\), is less than 1. This is an important concept because it allows the calculation of a finite sum even when the series consists of infinitely many terms.

Common Ratio

The common ratio is the factor by which we multiply each term to get the next term in a geometric series. For the given series, the common ratio \(r\) is 0.1. It plays a crucial role in determining the behavior of the series.

• If \(|r| < 1\), the series may converge, approaching a definite sum.
• If \(|r| = 1\), the series doesn't converge, as the terms don't diminish in size.
• If \(|r| > 1\), the series diverges, as the terms continue to grow.

Understanding the common ratio not only helps in identifying the nature of a geometric series but also in calculating the sum when possible. In our series, since \(r = 0.1\), it indicates that the infinite series will converge.
The capability to determine convergence or divergence, using the common ratio, allows for deeper exploration and application of series in solving practical mathematical problems.