Problem 16 - Advanced Engineering Mathematics

Question

In Problems 15-20, 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 First Term

For the given geometric series, the first term is the value of the series when \( k = 1 \). This is given by substituting \( k = 1 \) into the term formula: \( 4i \left(\frac{1}{3}\right)^{1-1} = 4i \times 1 = 4i \). Thus, the first term \( a = 4i \).

Step 2: Identify the Common Ratio

The common ratio \( r \) in a geometric series is the factor by which we multiply each term to get the next term. Here, \( r = \frac{1}{3} \), as observed from the term expression \( \left(\frac{1}{3}\right)^{k-1} \).

Step 3: Determine Convergence

A geometric series \( \sum_{k=1}^{ ext{∞}} ar^{k-1} \) converges if \(|r| < 1\). Here, \( |r| = \left|\frac{1}{3}\right| = \frac{1}{3} < 1\), so the series converges.

Step 4: Calculate the Sum of the Series

When a geometric series converges, its sum can be calculated using the formula \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio. Substituting the values, we have:\[ S = \frac{4i}{1 - \frac{1}{3}} = \frac{4i}{\frac{2}{3}} = 4i \times \frac{3}{2} = 6i \].

Step 5: Conclude the Solution

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

Key Concepts

Convergence of Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio. The convergence of a geometric series depends on the value of this common ratio. For a geometric series given by \( \sum_{k=1}^{\infty} ar^{k-1} \), it

• Converges if the absolute value of the common ratio \( r \) is less than 1, i.e., \( |r| < 1 \).
• Diverges if the absolute value of the common ratio is greater than or equal to 1.

In the problem we consider, the common ratio is \( \frac{1}{3} \) and its absolute value is \( \frac{1}{3} \). Since \( \frac{1}{3} < 1 \),
the series converges. Understanding this concept is crucial for determining the behavior of an infinite series. Without convergence, the sum of an infinite series cannot be defined.

Sum of Series

Once we establish that a geometric series converges, we can find its sum using a straightforward formula. For a convergent geometric series \( \sum_{k=1}^{\infty} ar^{k-1} \), the sum \( S \) is calculated as:

• \( S = \frac{a}{1 - r} \),

where \( a \) is the first term and \( r \) is the common ratio. In our given series, the first term \( a \) is \( 4i \) because when \( k = 1 \), the term is \( 4i \times 1 = 4i \). The common ratio \( r \) is \( \frac{1}{3} \). Thus, the sum is:
\[ S = \frac{4i}{1 - \frac{1}{3}} = \frac{4i}{\frac{2}{3}} = 6i. \]This formula makes it easy to find the sum without adding each term separately, which is especially helpful for infinite series. Knowing the series converges allows us to use this approach effectively.

Complex Numbers

Complex numbers extend our understanding of numbers by introducing the imaginary unit \( i \), which satisfies \( i^2 = -1 \). A complex number \( z \) can be expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit. In geometric series, like the one we are dealing with in the problem,

• the terms can involve complex numbers like \( 4i \).
• Multiplications and operations involving these terms are handled just like with regular numbers, keeping in mind the properties of \( i \).

For instance, when summing our series, we treat \( 4i \) just as we would any other number, applying the same convergence criteria and summing formulas. This blend of real and imaginary components allows complex numbers to solve problems that real numbers alone cannot address. They play a crucial role in various fields, including engineering and physics, where they help characterize waveforms and oscillations.