Problem 9 - Calculus: Early Transcendentals

Question

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why. \( \displaystyle \sum_{n = 1}^{\infty} \frac {12}{(-5)^n} \)

Step-by-Step Solution

Answer

Convergent; sum is -2.4.

Step 1: Understand the Series Formula

The series given is a geometric series: \( \sum_{n = 1}^{\infty} \frac {12}{(-5)^n} \). A geometric series has the form \( a + ar + ar^2 + ar^3 + \ldots \) where \( a \) is the first term and \( r \) is the common ratio.

Step 2: Identify Parameters

Here, the first term \( a = \frac{12}{-5} \) and the common ratio \( r = \frac{1}{-5} \). Check if \(|r| < 1\) to determine if the series converges.

Step 3: Determine Convergence or Divergence

Since the absolute value of the common ratio, \(|r| = \left| \frac{1}{-5} \right| = \frac{1}{5} < 1\), the series converges. Geometric series converge if and only if \(|r| < 1\).

Step 4: Calculate the Sum of the Series

The sum \( S \) of an infinite geometric series is given by the formula \( S = \frac{a}{1-r} \), where \( |r| < 1 \). For this series, \( S = \frac{\frac{12}{-5}}{1 - \frac{1}{-5}} = \frac{\frac{12}{-5}}{\frac{6}{5}} = -2.4 \).

Step 5: Calculate Initial Partial Sums

Calculate the first 10 partial sums to illustrate convergence. For example, the first partial sum \( S_1 = \frac{12}{-5} \), the second \( S_2 = \frac{12}{-5} + \frac{12}{25} \), and so on. Continue until \( S_{10} \).

Step 6: Graphing the Series and Partial Sums

Graph the sequence of terms \( a_n = \frac {12}{(-5)^n} \) to show a decreasing trend and graph the sequence of partial sums \( S_n \) to show the convergence to \(-2.4\). You should observe that the partial sums approach the sum of the series \(-2.4\).

Step 7: Conclusion on Convergence

The sequence of partial sums \( S_n \) visually and numerically approaches \(-2.4\), confirming convergence. The series converges and the sum is \(-2.4\).

Key Concepts

Partial Sums

In a geometric series, the concept of partial sums is crucial to understand convergence. A partial sum is the sum of the first few terms of a series. These are calculated to get insights into the behavior of the series. For the given geometric series, a partial sum can be represented as

• The first term: \( S_1 = a = \frac{12}{-5} \)
• The second partial sum: \( S_2 = a + ar \)
• Continuing up to \( S_n = a + ar + ar^2 + \cdots + ar^{n-1} \)

Calculating these sums helps illustrate whether the sums are stabilizing at a particular value, indicating convergence. In practice, plotting these values can display how the sequence of partial sums is moving closer to a potential sum, or warns us if it starts deviating instead.

Convergence and Divergence

A critical aspect in dealing with series is determining whether they converge or diverge. Convergence of a series implies that as you keep adding more terms, the total eventually settles around a finite value. For a geometric series, the condition for convergence is determined by the common ratio \( |r| < 1 \). In this exercise, the ratio is \( \left| \frac{1}{-5} \right| = \frac{1}{5} \), which is less than 1.

• If \( |r| < 1 \), the series converges, and there is a finite sum.
• If \( |r| \geq 1 \), the series diverges, meaning it does not have a finite closest value.

The series mentioned demonstrates convergence as its terms diminish in size and approach a particular value. If the partial sums repeatedly extending do not stabilize, the series is termed divergent.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. Although it sounds endless, such series can indeed have a finite sum if they converge. In our case, the \( \sum_{n=1}^{\infty} \frac{12}{(-5)^n} \) is an example of a geometric infinite series. While each term gets added, their reduced magnitude soon becomes negligible, allowing the overall sum to be finite.These series can be divided into two types:

• Finite length periodic series that cycle after some time.
• Genuinely infinite series, like the one exemplified, extending indefinitely.

In practice, calculating infinite series often involves identifying a pattern or formula, like in this exercise, to compute the sum efficiently rather than adding each term manually.

Sum of a Series

The sum of a geometric series, if convergent, can be efficiently calculated using a formula. For a geometric series, if the series converges, its sum can be derived as \[ S = \frac{a}{1 - r}\] where \( a \) is the first term and \( r \) is the common ratio. Applying this to our series:

• First term \( a = \frac{12}{-5} \)
• Common ratio \( r = \frac{1}{-5} \)
• Sum \( S = \frac{\frac{12}{-5}}{1 - \frac{1}{-5}} = -2.4 \)

This formula lets us quickly ascertain the series sum once convergence is confirmed, without having to sum thousands or millions of terms. Importantly, this methodology underlines the elegance of mathematics in dealing with seemingly eternal processes and distilling finite values from them.