Problem 15 - Advanced Engineering Mathematics

Question

Determine whether the given geometric series is convergent or divergent. If convergent, find its sum. \(\sum_{k=0}^{\infty}(1-i)^{k}\)

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Identify the First Term

The given series is \( \sum_{k=0}^{\infty}(1-i)^{k} \). The first term \( a \) of the series is the value of the expression when \( k=0 \). Thus, \( a = (1-i)^{0} = 1 \).

Step 2: Identify the Common Ratio

The common ratio \( r \) is the term that is used to multiply each term to get the next term in the series. It is given by what you multiply each term by: \( r = (1-i) \).

Step 3: Check for Convergence

A geometric series converges if the absolute value of the common ratio \( |r| \) is less than 1. Calculate \( |1-i| \):\[|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\]Since \( \sqrt{2} > 1 \), the series is divergent.

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are typically represented in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary part includes the imaginary unit \( i \), which is defined by the property \( i^2 = -1 \). This unique characteristic allows complex numbers to provide solutions to equations that would be unsolvable using only real numbers.
For instance, the complex number \( 1-i \) used in the original exercise consists of a real part (1) and an imaginary part (-1). When working with complex numbers, you often find yourself performing operations like addition, subtraction, multiplication, and division, just like with real numbers. In the step-by-step solution, they calculate the modulus or absolute value of \( 1-i \), which involves finding the distance of the number from the origin in a complex plane:

• Absolute value: \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\)

This modulus is crucial in determining properties such as convergence in a geometric series.

Convergence and Divergence

When analyzing series, one of the central concepts is determining whether the series is convergent or divergent. A geometric series is said to converge if the absolute value of its common ratio \( |r| \) is less than 1. This means the terms are getting smaller and approaching a limit, resulting in the series adding up to a finite value.
Conversely, a series diverges if \( |r| \geq 1 \). This means the terms of the series do not approach zero, causing the series to increase indefinitely or oscillate without approaching a specific value. In the original exercise, we determined that the series \( \sum_{k=0}^{\infty}(1-i)^k \) is divergent because:

• \(|1-i| = \sqrt{2}\), which is greater than 1.

Understanding convergence and divergence helps us predict the behavior of infinite series, allowing us to determine whether the series will result in a meaningful sum or not.

Infinite Series

An infinite series is the sum of an infinite sequence of terms. More formally, it is expressed in the form \( \sum_{k=0}^{\infty} a_k \), where \( a_k \) represents the sequence of terms.
This concept is particularly important in mathematics because it allows us to extend the ideas of summation beyond a finite number of terms to an infinite context. Geometric series are a fundamental example of infinite series. They follow the specific form \( a + ar + ar^2 + ar^3 + \ldots \) where \( a \) is the first term and \( r \) is the common ratio.
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• For a geometric series to have a sum, it must converge, which requires the criterion \( |r| < 1 \).

In many applications, especially in solving real-world problems, we use the concept of an infinite series to approximate functions and solve differential equations. It's fascinating to see how these series behave under different conditions, such as when the common ratio is a complex number, as seen in the exercise. The study of infinite series promises a deeper understanding of patterns and structures within mathematics.