Problem 15 - Advanced Engineering Mathematics

Question

In Problems 15-20, determine whether the given geometric series is convergent or divergent. If convergent, find its sum. $$ \sum_{k=0}^{\infty}(1-i)^{k} $$

Step-by-Step Solution

Answer

The series \( \sum_{k=0}^{\infty} (1-i)^{k} \) is divergent because \(|1-i| > 1\).

Step 1: Identify the series terms

We are given the geometric series \( \sum_{k=0}^{\infty} (1-i)^{k} \). This is an infinite series where the first term \( a = 1 \) and the common ratio \( r = 1-i \).

Step 2: Determine the convergence condition

A geometric series \( \sum_{k=0}^{\infty} ar^{k} \) converges if \( |r| < 1 \). We need to find the magnitude of the common ratio \( r = 1-i \).

Step 3: Calculate the magnitude of the common ratio

The magnitude of \( 1-i \) is calculated as follows: \[ |1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}. \] Since \( \sqrt{2} \approx 1.414 \, (> 1) \), the series does not meet the convergence condition.

Step 4: Conclusion about convergence

Since \( |r| > 1 \), the series \( \sum_{k=0}^{\infty} (1-i)^{k} \) is divergent. It does not converge to a finite sum.

Key Concepts

Convergence

In mathematical analysis, convergence refers to the behavior of a series or sequence. Specifically, a series is said to converge if the sequence of its partial sums approaches a finite limit. In the context of a geometric series, convergence depends on the value of the common ratio, denoted as \( r \).

For a geometric series \( \sum_{k=0}^{\infty} ar^{k} \), the series converges if the absolute value of the common ratio \(|r|\) is less than 1.

• If \(|r| < 1\), the series converges.
• If \(|r| \geq 1\), the series diverges.

This concept is crucial because a convergent series results in a finite sum, unlike a divergent series which does not settle to a particular value.

In practical terms, checking for convergence not only helps in determining whether a series can be summed to a particular value, but also aids in understanding the behavior of the series over an infinite breadth.

Divergence

Divergence, on the other hand, refers to the failure of a series to have a finite limit. For a geometric series, divergence occurs when the magnitude of the common ratio \(|r|\) is greater than or equal to 1. In such cases, the partial sums of the series continue to grow larger indefinitely.

This means that as you add more terms in the series, the sum does not approach any finite value. Instead, it tends to increase towards infinity, demonstrating fluctuating or unbounded behavior.

• For a geometric series to diverge, \(|r| \geq 1\).
• Divergent series cannot be assigned a finite sum.

Recognizing divergence is as valuable as recognizing convergence. It helps you decide when further investigation or transformation is needed because you cannot simply add up infinite series terms to get a meaningful result.

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part, typically expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) being the imaginary unit satisfying \( i^2 = -1 \).

In the context of the given series, the common ratio \( r = 1 - i \) is a complex number. When dealing with convergence and divergence, it's essential to understand the magnitude of this complex number, which is found by taking the square root of the sum of the squares of its real and imaginary components.

• The magnitude \(|a + bi| = \sqrt{a^2 + b^2}\).
• Complex numbers can significantly affect the convergence behavior of series.

Understanding complex numbers is vital because they show up in various mathematical settings, particularly with series involving imaginary parts, impacting how these series behave or sum up.