Problem 3 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Determine whether the given geometric series converges or diverges. If the series converges, find its sum. $$ \sum_{n=0}^{\infty} \frac{13}{(-5)^{n}} $$

Step-by-Step Solution

Answer

The series converges and its sum is \(\frac{65}{6}\).

Step 1: Identify the First Term and Common Ratio

The given series is \[ \sum_{n=0}^{\infty} \frac{13}{(-5)^{n}} \]. Here, the first term (\(a\)) is \(13\) and the common ratio (\(r\)) is \(\frac{1}{-5}\).

Step 2: Determine the Common Ratio

Calculate the common ratio for the series: \(r = \frac{1}{-5} = -\frac{1}{5}\)

Step 3: Check for Convergence Condition

A geometric series converges if the absolute value of the common ratio \(|r|\) is less than 1. Here, \(|r| = |-\frac{1}{5}| = \frac{1}{5} < 1\). Therefore, the series converges.

Step 4: Use the Formula for Sum of Infinite Geometric Series

For a convergent geometric series, the sum is given by: \(S = \frac{a}{1-r}\). Substitute the values of \(a\) and \(r\) from Step 1: \[S = \frac{13}{1 - (-\frac{1}{5})} = \frac{13}{1 + \frac{1}{5}} = \frac{13}{\frac{6}{5}} = \frac{13 \cdot 5}{6} = \frac{65}{6}\]

Key Concepts

geometric series

A geometric series is characterized by each term being a constant multiple of the previous term. In this type of series, there are two main parts:

• The first term (denoted as \(a\))
• The common ratio (denoted as \(r\))

For example, in the series \(\sum_{n=0}^{\infty} \frac{13}{(-5)^{n}} \), the first term is 13, and the common ratio is \(-\frac{1}{5}\).

Each term in a geometric series is found by multiplying the previous term by the common ratio.
If you start with the first term as 13, the next terms will be: \(13, 13 \cdot \left( -\frac{1}{5} \right) , 13 \cdot \left( -\frac{1}{5} \right)^2\), and so on.
We can describe the general form of a geometric series as:

\[ \sum_{n=0}^{\infty} ar^{n} \], where:

• \(a\) is the first term
• \(r\) is the common ratio

convergence criteria

Convergence is a key concept when dealing with infinite series. For a geometric series to converge, the absolute value of the common ratio must be less than 1.

In other words, \(|r| < 1 \).

Convergence means that as the number of terms increases indefinitely, the sum of the series approaches a specific finite value.
This happens when each incremental term added to the series gets smaller and smaller.

For the series \(sum_{n=0}^{\infty} \frac{13}{(-5)^{n}} \), the common ratio is \( -\frac{1}{5}\), and its absolute value is: \(|r| = \left| -\frac{1}{5}\right| = \frac{1}{5} < 1\).

This meets the convergence criterion. Therefore, the series converges.

sum of infinite series

Once a geometric series is determined to converge, we can find its sum using a specific formula.
The formula to find the sum of an infinite geometric series is:

\[ S = \frac{a}{1-r} \],
where:

• \(S\) is the sum
• \(a\) is the first term
• \(r\) is the common ratio

In the example series \(\sum_{n=0}^{\infty} \frac{13}{(-5)^{n}} \), we already calculated our first term \(13\) and our common ratio as \(-\frac{1}{5}\).

Plugging these values into the formula, we get:
\[S = \frac{13}{1 - \left( -\frac{1}{5} \right)} = \frac{13}{1 + \frac{1}{5}} = \frac{13}{\frac{6}{5}} = \frac{13 \cdot 5}{6} = \frac{65}{6} \]

So, the sum of the infinite series is \(\frac{65}{6}\).
This process helps in finding the value that the infinite series approaches, ensuring the concepts of convergence and sum are well understood.