Problem 18 - A First Course in Complex Analysis with Applications

Question

Determine whether the given geometric series is convergent or divergent. If convergent, find its sum. $$ \sum_{k=0}^{\infty} \frac{1}{2} i^{k} $$

Step-by-Step Solution

Answer

The series is divergent because the absolute value of the common ratio is 1.

Step 1: Recognize the Series

The series given is \( \sum_{k=0}^{\infty} \frac{1}{2} i^{k} \). This is a geometric series where each term is obtained by multiplying the previous term by a fixed ratio.

Step 2: Identify the First Term

The first term of the series can be identified when \( k = 0 \). So, the first term \( a = \frac{1}{2} i^0 = \frac{1}{2} \cdot 1 = \frac{1}{2} \).

Step 3: Determine the Common Ratio

The common ratio \( r \) is the factor you multiply each term by to get the next term. For this series, \( r = i \), since for each term, \( i \) is raised to the next power.

Step 4: Check Convergence Condition

A geometric series is convergent if the absolute value of the common ratio \( |r| < 1 \). In this case, \( r = i \), so \( |i| = 1 \). Since \( |r| = 1 \) and not less than 1, the series is divergent.

Key Concepts

Convergent Series

A geometric series is a special type of series where each term is found by multiplying the previous term by a constant, known as the common ratio. One crucial aspect of a geometric series is whether it converges or diverges. For a geometric series to be convergent, the absolute value of its common ratio must be less than one. Mathematically, this is expressed as \[|r| < 1\] If the condition is met, the series approaches a specific sum as more terms are added. Geometric convergent series are important because they allow us to find finite sums for series that would otherwise go on infinitely. They have applications in various fields, such as physics, engineering, and finance, where predicting outcomes using series is necessary. However, if the absolute value of the common ratio is equal to or greater than one, the series does not converge to a sum but instead is divergent.

Divergent Series

Divergent series, unlike convergent series, do not settle on a particular sum as you add up terms indefinitely. They are characterized by the fact that their terms do not approach zero fast enough, or the progression grows without bound. In geometric series, divergence occurs when the absolute value of the common ratio is equal to or exceeds one:

• If \(|r| = 1\), any geometric series will alternate or cycle through values without reaching a sum.
• If \(|r| > 1\), the series will grow endlessly, leading the total to infinity.

Understanding divergent series is vital in calculus and analysis because it helps students know when summation will lead to lack of results. In the exercise provided, because the common ratio is a complex number with a modulus of one, it confirms the series is divergent.

Complex Numbers

Complex numbers expand the traditional understanding of numbers beyond just real numbers. They consist of a real part and an imaginary part, usually denoted as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property \(i^2 = -1\). This property allows for new dimensions of problem solving especially in fields like engineering and physics.In the context of geometric series, as seen in the exercise, the common ratio is the complex number \(i\). The absolute value (or modulus) of a complex number \(a + bi\) is calculated using:\[|a + bi| = \sqrt{a^2 + b^2}\]For \(i\), the calculation is \(|i| = \sqrt{0 + 1} = 1\). This modulus being equal to one makes the geometric series in the given exercise divergent.Complex numbers have broad applications, including solving polynomial equations (like square roots of negative numbers), electrical engineering (analyzing AC circuits), and in quantum mechanics. These numbers thus enable us to address more sophisticated and real-world problems.