Problem 16 - Advanced Engineering Mathematics

Question

Determine whether the given geometric series is convergent or divergent. If convergent, find its sum. \(\sum_{k=1}^{\infty} 4 i\left(\frac{1}{3}\right)^{k-1}\)

Step-by-Step Solution

Answer

The series is convergent and its sum is \( 6i \).

Step 1: Identify the Geometric Series

The series is given as \( \sum_{k=1}^{\infty} 4i \left( \frac{1}{3} \right)^{k-1} \). This is a geometric series where the first term \( a = 4i \) and the common ratio \( r = \frac{1}{3} \).

Step 2: Check the Convergence of the Series

A geometric series \( \sum_{k=0}^{\infty} ar^k \) converges if the absolute value of the common ratio \( |r| < 1 \). Here, the common ratio \( r = \frac{1}{3} \) and \( \left| \frac{1}{3} \right| < 1 \), so the series is convergent.

Step 3: Find the Sum of the Convergent Series

For a convergent geometric series \( \sum_{k=0}^{\infty} ar^k \), the sum is given by \( S = \frac{a}{1-r} \). Here, \( a = 4i \) and \( r = \frac{1}{3} \). Substituting these values gives \( S = \frac{4i}{1 - \frac{1}{3}} = \frac{4i}{\frac{2}{3}} = 6i \).

Step 4: Final Verification

Ensure the solution is consistent by verifying the calculations. We used the correct formulas and calculated \( S \) correctly as \( S = \frac{4i \times 3}{2} = 6i \). The series is convergent and its sum is \( 6i \).

Key Concepts

Convergence

In mathematics, convergence is a key concept when examining infinite series such as geometric series. We say that a series is convergent if it approaches a finite limit as more terms are added. The idea of convergence helps us understand whether adding infinite terms results in a finite sum.

For a geometric series to converge, the absolute value of the common ratio must be less than one. This ensures that the terms get progressively smaller, allowing the overall sum to stabilize and reach a limit. In our example, the common ratio is \( r = \frac{1}{3} \), which satisfies \( |r| < 1 \). This indicates that our series converges, leading us to a finite sum.

Convergence criteria are crucial as they provide the essential checks on whether you can find a sum for an otherwise infinite process.

Complex Numbers

Complex numbers expand the realm of real numbers, introducing a new dimension that includes both real and imaginary parts. Denoted as \( a + bi \), where \( i \) is the imaginary unit satisfying the equation \( i^2 = -1 \), complex numbers are vital in many areas of mathematics and engineering.

In the given geometric series, the first term is \( 4i \). This shows how complex numbers can be part of practical mathematical evaluations. Here, \( 4i \) serves as the initial value around which the series builds and expands based on the convergence rules mentioned.

Handling complex numbers often involves managing their real and imaginary components separately. They are integrated seamlessly into formulas like those for geometric series, when finding sums or assessing convergence.

Infinite Series

An infinite series is the sum of an unending sequence of terms. These series are core to calculus and many higher mathematical concepts. Unlike finite series with a limited number of terms, infinite series stretch this notion by attempting to analyze the behavior of sums approaching infinity.

Geometric series, like the one in our problem, are a type of infinite series where each term is a constant multiple (the common ratio) of the previous term. The series can either converge, which is when it approaches a specific value, or diverge when it grows without bounds.

Infinite series are often used in analytic contexts to represent functions, solve equations, or approximate values in scientific computations. Understanding their behavior and convergence is crucial in applying infinite series effectively across different disciplines.