Problem 18 - Advanced Engineering Mathematics

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; it does not converge.

Step 1: Understand the General Form of a Geometric Series

A geometric series is of the form \(\sum_{k=0}^{\infty} ar^k\), where \(a\) is the first term and \(r\) is the common ratio. In this series, \(a=\frac{1}{2}\) and \(r=i\).

Step 2: Determine Convergence Criteria

A geometric series converges if the absolute value of the common ratio \(|r| < 1\). Here, \(|i| = 1\), since the magnitude of \(i\) (a complex number) is 1.

Step 3: Apply Convergence Test

Since \(|r| = 1\), the series does not meet the convergence criterion \(|r| < 1\). Therefore, the series is divergent.

Key Concepts

Convergence Criteria

When tackling geometric series, the first thing we need to determine is whether the series converges or diverges. This fundamentally boils down to understanding the convergence criteria for geometric series. A geometric series is expressed as \( \sum_{k=0}^{\infty} ar^k \), where \( a \) is the first term and \( r \) is the common ratio. The critical condition for convergence is the absolute value of the common ratio, denoted by \( |r| \), must be less than 1.

Here's why this matters:

• If \( |r| < 1 \), the terms of the series get smaller and approach zero as \( k \) approaches infinity. This makes the series convergent, allowing us to sum it to an actual finite number.
• If \( |r| \geq 1 \), the terms do not shrink enough for convergence. Hence, the series diverges, meaning it can grow infinitely or oscillate without approaching a specific number.

In the given series \( \sum_{k=0}^{\infty} \frac{1}{2} i^{k} \), the common ratio \( r = i \) has a magnitude \( |i| = 1 \). Using the convergence criteria, since \( |i| ot< 1 \), the series cannot converge. Understanding convergence criteria is vital for analyzing any geometric series.

Complex Numbers

Complex numbers like \( i \) are crucial in analyzing series that include imaginary components. The number \( i \) represents the square root of -1, and it plays a fundamental role beyond simple real number arithmetic. A complex number is often represented as \( a + bi \), where \( a \) and \( b \) are real numbers.

When dealing with geometric series that have a complex ratio, the magnitude (or modulus) of the complex number becomes significant. The magnitude for a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \). For \( i \) specifically (where \( a = 0 \) and \( b = 1 \)), the magnitude is \( |i| = \sqrt{0^2 + 1^2} = 1 \). Understanding how to interpret the magnitude is essential because it directly informs us about whether the series will converge or diverge.

The involvement of complex numbers can make the task much more interesting. However, it requires a solid understanding of both the arithmetic and algebraic properties of complex numbers to effectively determine the behavior of geometric series with complex ratios. Without this understanding, one might struggle to comprehend series like the one in the exercise.

Series Divergence

Divergence in series is a concept that signals the inability of a series to settle at a finite sum. When a geometric series does not meet the necessary convergence criteria, it diverges.

There are several telltale signs for divergence:

• The absolute value of the common ratio \( |r| \) is greater than or equal to 1, meaning the terms do not adequately decrease in magnitude.
• Specifically in the given series \( \sum_{k=0}^{\infty} \frac{1}{2} i^{k} \), the ratio \( r = i \) has a magnitude of 1 (\( |i| = 1 \)), which is precisely on the boundary of convergence.

Evaluating a series that diverges:

• The terms start repeating or magnifying because they do not approach zero.
• Such series can also grow infinitely or oscillate without settling to a fixed number.

The recognition of a divergent series is crucial as it indicates that further analysis to find a sum is futile. Instead of focusing on a nonexistent sum, one must understand and explain why divergence occurs. This understanding prevents confusion and unnecessary calculations that attempt to achieve convergence where it is impossible.