Problem 17 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Convergent; sum is \( \frac{2i - 1}{5} \).

Step 1: Identify the Common Ratio

For the series \( \sum_{k=1}^{\infty} \left(\frac{i}{2}\right)^{k} \), the term is \( a_k = \left(\frac{i}{2}\right)^{k} \). The common ratio \( r \) for a geometric series is the expression raised to the power \( k \). Here, \( r = \frac{i}{2} \).

Step 2: Determine Convergence Criteria

A geometric series converges only when the absolute value of the common ratio \( |r| \) is less than 1. Compute the absolute value: \( |r| = \left|\frac{i}{2}\right| = \frac{1}{2} \), since \( |i| = 1 \). Therefore, because \( \frac{1}{2} < 1 \), the series converges.

Step 3: Find the Sum of the Series

For a geometric series where \( |r| < 1 \), the sum can be calculated using \( S = \frac{a}{1-r} \), where \( a \) is the first term of the series. Here, \( a = \left(\frac{i}{2}\right)^{1} = \frac{i}{2} \). Thus, the sum \( S \) is given by:\[S = \frac{\frac{i}{2}}{1 - \frac{i}{2}} = \frac{\frac{i}{2}}{\frac{2-i}{2}} = \frac{i}{2-i}.\]

Step 4: Simplify the Sum Expression

To simplify \( \frac{i}{2-i} \), multiply the numerator and the denominator by the conjugate of the denominator, \( 2+i \):\[\frac{i(2+i)}{(2-i)(2+i)} = \frac{2i - 1}{5} = \frac{2i - 1}{5}.\]Therefore, the sum of the series is \( \frac{2i - 1}{5} \).

Key Concepts

Understanding Series Convergence

A key concept in discussing geometric series is whether the series will converge or not. Convergence in a series means that as you add more and more terms, the series approaches a specific value rather than just infinitely increasing. For geometric series, the crucial factor determining convergence is the "common ratio".

• If the absolute value of this common ratio is less than 1, the series converges.
• If the absolute value is 1 or greater, the series diverges, meaning it does not settle at a particular value.

It's important to calculate the absolute value of the common ratio correctly to determine convergence. This often serves as the first step in analyzing any geometric series. In the given example, the common ratio is \(\frac{i}{2}\), and calculating \(|r| = \left|\frac{i}{2}\right| = \frac{1}{2}\), helps us confirm that the series converges because \(\frac{1}{2} < 1\). Understanding this criterion is fundamental before moving on to find the series sum.

The Role of the Common Ratio

The common ratio is a vital component of any geometric series. It is essentially the factor by which each term in the series is multiplied to get the subsequent term. This constant ratio is what defines the series as geometric.

For the series \( \sum_{k=1}^{\infty} \left(\frac{i}{2}\right)^{k} \), the common ratio \(r\) is \(\frac{i}{2}\). Complex numbers like \(i\) often appear in advanced problems and can initially seem challenging. However, they follow the same rules for convergence when arranged in a series.

Knowing how to effectively determine and manipulate the common ratio can greatly influence your ability to analyze a geometric series. This geometric series progresses by multiplying each term by \(\frac{i}{2}\). Understanding how to handle complex numbers here, particularly their absolute value, allows you to assess convergence correctly.

Exploring Complex Numbers in Series

Complex numbers, such as \(i\), bring a unique twist to geometric series. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This definition can influence how terms in the series behave.

When dealing with complex numbers in series, it is important to remember their effect on calculations like finding the absolute value. The absolute value of a complex number \(a + bi\) is calculated as \(\sqrt{a^2 + b^2}\). For \(i\), which is equivalent to \(0 + 1i\), the absolute value is 1, as calculated by \(\sqrt{0^2 + 1^2} = 1\).

• Handling complex components involves basic complex arithmetic rules.
• Operations like finding sums often require multiplying by conjugates, as seen in the solution \(\frac{i}{2-i}\).

This step is crucial for simplifying and thereby correctly interpreting series sums. In addressing the series \(\sum_{k=1}^{\infty}\left(\frac{i}{2}\right)^{k}\), the adjustment with the conjugate \(2+i\) simplifies the series to a manageable expression, ensuring a clear understanding of its behavior and properties.