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

Question

For each series, determine whether the series converges absolutely, converges conditionally, or diverges. $$ \sum_{k=0}^{\infty}\left(-\frac{11}{12}\right)^{k} $$

Step-by-Step Solution

Answer

The series \(\sum_{k=0}^{\infty}\left(-\frac{11}{12}\right)^{k}\) is a geometric series that converges absolutely because the absolute value of the common ratio is less than 1.

Step 1: Identify the Series

Identify the series as a geometric series. A geometric series has a common ratio between successive terms. In this series, the common ratio is \(-\frac{11}{12}\)

Step 2: Check the common ratio

Determine the common ratio of the series. The formula for a geometric series is \[ a + ar + ar^{2} + .... + ar^{n} + ... \]. In our series, a = 1 and r = -\frac{11}{12}, which means we have a common ratio of -\frac{11}{12}. So, only the sign alternates with each term, but the magnitude remains the same, \(\frac{11}{12}\)

Step 3: Apply the Convergence test

For a geometric series \(\sum a_{n}= a*r^{(n-1)}\) the series converges if and only if \(|r| < 1\). In this series \(|r| = |\(-\frac{11}{12}\)| = \(\frac{11}{12}\). Consequently, since \(\frac{11}{12} < 1\), then the series converges absolutely.

Key Concepts

Geometric Series

A geometric series is a series of numbers with a constant ratio, called the common ratio, between consecutive terms. It usually takes the form:

• \( a + ar + ar^2 + ar^3 + \cdots \)

In this series, \( a \) is the first term and \( r \) is the common ratio. A well-known property of geometric series is that they can either converge or diverge, depending on the value of the common ratio \( r \).
For convergence, \(|r| < 1\); if this condition is not met, the series diverges.
To identify a geometric series, ensure that the ratio between successive terms remains unchanged. This makes it easier to analyze and predict the series' behavior.
The series \( \sum_{k=0}^{\infty}\left(-\frac{11}{12}\right)^{k} \) is a geometric series where \( a = 1 \) and \( r = -\frac{11}{12} \). The alternating sign does not affect the magnitude, only the direction of terms, ensuring that \( r \) remains constant.

Absolute Convergence

Absolute convergence is an important concept in determining the behavior of series. A series \( \sum a_n \) is said to converge absolutely if the series of absolute values \( \sum |a_n| \) also converges. This implies that if a series converges absolutely, it must converge in the ordinary sense as well.
For our geometric series example, the absolute value of the common ratio \( r = -\frac{11}{12} \) is \( \frac{11}{12} \). Because this value is less than 1, the series converges absolutely. This means that no matter how the terms fluctuate in terms of sign, the magnitude of terms is shrinking such that the sum approaches a finite limit.
In essence, absolute convergence is a stronger form of convergence, ensuring that the series behaves well even under rearrangement of its terms.

Convergence Tests

Convergence tests are crucial tools in analyzing whether a series converges or diverges. Various tests can be applied depending on the nature of the series.

• Ratio Test: This is especially useful for geometric series. It checks if the absolute value of the common ratio \( |r| \) is less than 1, indicating convergence.
• Comparison Test: Used to compare a series with a known convergent or divergent series to establish the likely behavior of the series in question.
• Root Test: Applicable when terms of the series involve powers, checking if the limit of the nth root of the absolute value of terms is less than 1.

For the series \( \sum_{k=0}^{\infty}\left(-\frac{11}{12}\right)^{k} \), the Ratio Test is applicable, and since \( |r| = \frac{11}{12} < 1 \), the series converges.
Applying these tests aids in systematically understanding diverse series, providing a clear path to determine whether a series will finally settle or diverge.