Problem 15 - A First Course in Complex Analysis with Applications

Question

In Problems 15-20, 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 diverges.

Step 1: Identify the form of the series

The series is given as \( \sum_{k=0}^{\infty}(1-i)^{k} \). This is a geometric series of the form \( a + ar + ar^2 + \ldots \), where \( a \) is the first term and \( r \) is the common ratio. In this series, \( a = 1 \) and \( r = 1-i \).

Step 2: Check the condition for convergence

A geometric series \( \sum_{k=0}^{\infty} ar^k \) converges if and only if the absolute value of the common ratio \( |r| < 1 \). Our common ratio is \( r = 1-i \).

Step 3: Calculate the magnitude of the common ratio

To find \(|1-i|\), calculate the magnitude:\[|1-i| = \sqrt{(1)^2 + (-1)^2} = \sqrt{2}\]Since \(\sqrt{2} > 1\), the magnitude of \(1-i\) is greater than 1.

Step 4: Determine the convergence or divergence

Since \(|1-i| = \sqrt{2} > 1\), the series \( \sum_{k=0}^{\infty}(1-i)^{k} \) does not satisfy the convergence condition for a geometric series. Thus, the series diverges.

Key Concepts

Convergence of Series

When exploring the convergence of a series, it's important to understand when a series will settle at a specific value or continue endlessly. In mathematics, a series converges if the sum of its infinite terms approaches a fixed value as more terms are added. For a geometric series, which is a series of numbers where each term is a fixed multiple of the previous one, convergence depends on the common ratio, denoted as \( r \).

The essential rule for convergence is straightforward: a geometric series \( \sum_{k=0}^{\infty} ar^k \) converges if the absolute value of the common ratio \( |r| \) is less than 1. This means that as k increases, the terms \( ar^k \) become smaller and smaller in absolute value, allowing the overall sum to approach a finite limit.

• If \( |r| < 1 \), the series converges, and you can find the sum using the formula \( \frac{a}{1-r} \), where \( a \) is the first term of the series.
• If \( |r| \geq 1 \), the series does not converge, meaning it is divergent.

Understanding these conditions helps in analyzing the behavior and determining the solutions to series problems.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. The standard form is \( a + bi \), where \( a \) is the real component and \( bi \) is the imaginary component. Imaginary numbers use \( i \), which is the square root of -1.

When working with series, complex numbers introduce unique conditions for analyzing convergence or divergence. In geometric series, the common ratio \( r \) can be a complex number. The convergence criteria still apply, but you need to consider the magnitude (or absolute value) of the complex number.

The magnitude of a complex number \( a + bi \) is calculated using the formula \( |a + bi| = \sqrt{a^2 + b^2} \). This helps in determining if a series will converge when such numbers are involved. For instance, in the series \( \sum_{k=0}^{\infty} (1-i)^k \), the common ratio \( r = 1-i \) has a magnitude that influences the convergence analysis.

Divergent Series

A series is said to be divergent if it does not converge to a limit as more terms are added. In mathematical terms, a divergent series will have terms that either grow without bound or do not settle to a definite number.

For geometric series, the condition for divergence is tied directly to the value of the common ratio \( r \). If the magnitude \( |r| \geq 1 \), the series is divergent. This means the terms are not decreasing in a manner that they sum to a finite value, leading the series to drift towards infinity or oscillate.

• This characteristic affects both real and complex numbers in the series.
• In our example, the series \( \sum_{k=0}^{\infty}(1-i)^k \) is divergent because the common ratio \( |1-i| = \sqrt{2} > 1 \).
• Since it does not meet the convergence condition, the sum cannot be determined.

Recognizing a divergent series is key in understanding whether it is worthwhile to try to find a sum or conclude that it cannot be summed finitely.