Problem 6 - Algebra 2 and Trigonometry

Question

a. Write each series in sigma notation. b. Determine whether each sum increases without limit, decreases without limit, or approaches a finite limit. If the series has a finite limit, find that limit. \(5+1+\frac{1}{5}+\frac{1}{25}+\cdots\)

Step-by-Step Solution

Answer

The series is \(\sum_{n=1}^{\infty} 5 \left(\frac{1}{5}\right)^{n-1}\) and it converges to \(\frac{25}{4}\).

Step 1: Identify the series pattern

The series provided is: \(5 + 1 + \frac{1}{5} + \frac{1}{25} + \cdots\). Notice that it is a geometric series with the first term \(a = 5\) and a common ratio \(r = \frac{1}{5}\). Each term is the previous term multiplied by \(\frac{1}{5}\).

Step 2: Express the series in sigma notation

The general term of a geometric series can be expressed as \( a \cdot r^{n-1} \). In this problem, \( a = 5 \) and \( r = \frac{1}{5} \). Thus, the series in sigma notation is:\[\sum_{n=1}^{\infty} 5 \cdot \left( \frac{1}{5} \right)^{n-1}.\]

Step 3: Determine the nature of the series

Since this is a geometric series with a common ratio \( r = \frac{1}{5} < 1 \), the series converges. Geometric series converge to a finite limit when \(|r| < 1\).

Step 4: Calculate the finite limit of the series

The sum of an infinite geometric series is calculated using the formula \( S = \frac{a}{1-r} \). Here, \( a = 5 \) and \( r = \frac{1}{5} \), so the series converges to:\[S = \frac{5}{1 - \frac{1}{5}} = \frac{5}{\frac{4}{5}} = \frac{5 \times 5}{4} = \frac{25}{4}.\]

Key Concepts

Sigma Notation

In mathematics, sigma notation is a way to represent a series using a standard and concise format. It allows the expression of a sequence of terms that are added together, giving us an efficient method to denote long sums. In the context of geometric series, sigma notation helps us express the sum of terms that follow a specific pattern easily.

For example, the series given in the exercise is:

• 5
• 1
• \( \frac{1}{5} \)
• \( \frac{1}{25} \)
• ...

These terms follow a geometric pattern with the first term \( a = 5 \) and common ratio \( r = \frac{1}{5} \). When expressing it in sigma notation, we use the formula for a geometric series: \( a \cdot r^{n-1} \), which in this case changes to \( 5 \cdot \left( \frac{1}{5} \right)^{n-1} \).

Using sigma notation, the series is represented as:

• \[ \sum_{n=1}^{\infty} 5 \cdot \left( \frac{1}{5} \right)^{n-1} \]

This notation lets us quickly see the series' start point and the rule for generating additional terms. It's a powerful tool for dealing with infinite series, especially in calculus and higher-level mathematics.

Series Convergence

Series convergence refers to the behavior of an infinite series as more terms are added. Specifically, a series is said to converge if the sequence of its partial sums approaches a specific, finite value. Conversely, if the sums grow indefinitely either positively or negatively, the series is said to diverge.

In geometric series, convergence is largely determined by the common ratio \( r \). The series will converge if the absolute value of the common ratio is less than 1. Mathematically, this can be described as

• \(|r| < 1 \)

In the exercise provided, the series has a common ratio of \( \frac{1}{5} \). Since this value is less than 1, the series is convergent. It means that as we continue to add more terms from this series, the overall sum will approach a certain finite limit rather than growing indefinitely.

This concept of convergence is essential in calculus and helps in understanding the behavior of functions and sequences. It is also important in practical applications like signal processing and financial analysis.

Finite Limit

When an infinite series converges, it approaches a finite limit, which is a specific value the series tends to as the number of terms grows. Knowing this limit helps in various mathematical and real-world applications, as it provides a concrete value from an otherwise infinite process.

To find the finite limit of a convergent geometric series, we use the formula:

• \[ S = \frac{a}{1-r} \]

where \( a \) is the first term, and \( r \) is the common ratio. For the series in the exercise, \( a = 5 \) and \( r = \frac{1}{5} \). Plugging these values into the formula gives:

\[ S = \frac{5}{1 - \frac{1}{5}} = \frac{5}{\frac{4}{5}} = \frac{25}{4} \]

This result, \( \frac{25}{4} \), is the finite limit of the series. This calculation shows how even an infinite series can sum to a well-defined number, illustrating the beauty and power of mathematical analysis.