Problem 25 - Advanced Engineering Mathematics

Question

Find the circle and radius of convergence of the given power series. \(\sum_{k=0}^{\infty}(1+3 i)^{k}(z-i)^{k}\)

Step-by-Step Solution

Answer

The circle of convergence is centered at \( z = i \) with radius \( \frac{1}{\sqrt{10}} \).

Step 1: Identify the Power Series

The given power series is \( \sum_{k=0}^{\infty}(1+3i)^{k}(z-i)^{k} \). This series has the form \( \sum_{k=0}^{\infty} a_k (z - z_0)^k \), where \( a_k = (1+3i)^k \) and \( z_0 = i \).

Step 2: Find the Center of the Circle of Convergence

For a power series \( \sum_{k=0}^{\infty} a_k (z - z_0)^k \), the center of the circle of convergence is given by the value of \( z_0 \). For this series, \( z_0 = i \). Thus, the center of convergence is at \( z = i \).

Step 3: Use the Formula for Radius of Convergence

The radius of convergence \( R \) of a power series \( \sum_{k=0}^{\infty} a_k (z - z_0)^k \) can be found using the formula: \( \frac{1}{R} = \limsup_{k \to \infty} \sqrt[k]{|a_k|} \). Here, \( a_k = (1+3i)^k \), so we need to evaluate \( \sqrt[k]{|(1+3i)^k|} \).

Step 4: Calculate the Modulus of the Coefficients

First, find \( |a_k| = |(1+3i)^k| = |1+3i|^k \). Calculate \( |1+3i| = \sqrt{1^2 + 3^2} = \sqrt{10} \). Therefore, \( |a_k| = (\sqrt{10})^k \).

Step 5: Determine the Radius of Convergence

Now calculate \( \limsup_{k \to \infty} \sqrt[k]{|(\sqrt{10})^k|} = \limsup_{k \to \infty} \sqrt[k]{10^{k/2}} = \limsup_{k \to \infty} 10^{1/2} = \sqrt{10} \). Therefore, \( \frac{1}{R} = \sqrt{10} \), which gives \( R = \frac{1}{\sqrt{10}} \).

Step 6: State the Circle and Radius of Convergence

The circle of convergence has its center at \( z = i \) and a radius of \( \frac{1}{\sqrt{10}} \). The power series converges within this circle.

Key Concepts

Power Series

In mathematics, a power series is an infinite series of the form \( \sum_{k=0}^{\infty} a_k (z - z_0)^k \), where each term is a product of a coefficient \( a_k \) and a power of \((z - z_0)\). Here, \(z_0\) is a constant that represents the center of the series. Power series play a critical role in both real and complex analysis.

• They are useful for approximating functions and solving differential equations.
• Power series have applications in various fields, such as physics and engineering, where they model phenomena like heat and wave equations.
• They converge within a certain radius, beyond which the series does not converge.

Understanding power series involves not only grasping how they are constructed but also the conditions under which they converge, which brings us to the concept of the radius and circle of convergence. By adjusting the variable \(z\), we can examine how the series behaves as it moves away from the center \(z_0\). The magnitude of this behavior is known as the radius of convergence, a measure that determines the boundary between convergence and divergence.

Complex Numbers

Complex numbers extend real numbers by incorporating the imaginary unit \(i\), where \(i^2 = -1\). They have the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. In the context of power series, complex numbers allow for the exploration of series not just along the real line but across the entire complex plane.

• Complex numbers provide the fundamental framework for complex analysis.
• They are used to describe vibrations, electrical circuits, and quantum mechanics.
• The modulus \(|a + bi| = \sqrt{a^2 + b^2}\) measures the distance from the origin to the point \((a, b)\) in the complex plane.

When dealing with a power series involving complex numbers, like \((1 + 3i)^k\), computational focus often turns to the modulus of the coefficients to determine convergence properties. This modulus plays a fundamental role in evaluating the radius of convergence of the series as seen in exercises involving power series.

Circle of Convergence

The circle of convergence is a fundamental concept related to power series, specifically in the context of complex numbers. For a power series centered at \(z_0\), the circle of convergence is the region within which the series converges. This region takes the form of a disk in the complex plane.

• The disk is centered at \(z_0\), the center of the series.
• The radius of the disk is the radius of convergence, \(R\).
• Determining \(R\) involves computing the lim sup of the \(k\)th root of the modulus of the series coefficients.

For example, in \(\sum_{k=0}^{\infty}(1+3i)^{k}(z-i)^{k}\), the center \(z_0\) is \(i\), and the radius of convergence is calculated as \(1/\sqrt{10}\). This means the power series converges for all values of \(z\) within this distance from the center. Therefore, understanding the circle of convergence is key to determining where the power series will provide useful approximations and where it ceases to be effective.