Problem 24 - Advanced Engineering Mathematics

Question

In Problems 21-28, find the circle and radius of convergence of the given power series. $$ \sum_{k=1}^{\infty} \frac{1}{k^{2}(3+4 i)^{k}}(z+3 i)^{k} $$

Step-by-Step Solution

Answer

The series converges for \( |z + 3i| < 5 \), with the center \( z = -3i \).

Step 1: Identify the Form of the Power Series

The given power series is \( \sum_{k=1}^{\infty} \frac{1}{k^{2}(3+4i)^{k}}(z+3i)^{k} \). This is a power series of the form \( \sum_{k=0}^{\infty} a_k (z - z_0)^k \) where \( a_k = \frac{1}{k^2 (3+4i)^k} \) and \( z_0 = -3i \). Here \( z_0 \) is the center of the circle of convergence.

Step 2: Use the Formula for Radius of Convergence

The radius of convergence \( R \) of a power series \( \sum a_k (z - z_0)^k \) can often be found using the formula \( \frac{1}{R} = \limsup_{k \to \infty} \sqrt[k]{|a_k|} \). In our case, \( a_k = \frac{1}{k^2 (3+4i)^k} \).

Step 3: Simplify the \( a_k \) Terms

First, calculate the modulus of \( a_k \): \[|a_k| = \left| \frac{1}{k^2 (3+4i)^k} \right| = \frac{1}{k^2 |3+4i|^k}\]The modulus \( |3+4i| \) is calculated as:\[|3+4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.\]So, \(|a_k| = \frac{1}{k^2 \cdot 5^k}\).

Step 4: Calculate the Limit Superior

Next, we find the \( \limsup \) of \( |a_k| \) raised to the power \( \frac{1}{k} \):\[\limsup_{k \to \infty} \left( \frac{1}{k^2 \cdot 5^k} \right)^{1/k} = \limsup_{k \to \infty} \frac{1}{k^{2/k} \cdot 5} = \frac{1}{5}\]Here, the factor \( k^{2/k} \to 1 \) as \( k \to \infty \).

Step 5: Determine the Radius of Convergence

From the previous step, we have \( \limsup_{k \to \infty} \sqrt[k]{|a_k|} = \frac{1}{5} \). Therefore, using the relation \( \frac{1}{R} = \frac{1}{5} \), we find the radius of convergence \( R = 5 \).

Step 6: Conclusion

The circle of convergence for the given power series is centered at \( z = -3i \) with a radius of \( 5 \). This means the series converges for all \( z \) such that \( |z + 3i| < 5 \).

Key Concepts

Complex Power Series

A **complex power series** is a series of the form \( \sum_{k=0}^{\infty} a_k(z - z_0)^k \), where \( z \) is a complex variable, \( z_0 \) is the center of the series, and \( a_k \) are complex coefficients. These series can be thought of as a way to represent complex functions as infinite sums.

• The series converges at some points in the complex plane, which depends on the nature of the coefficients \( a_k \).
• Finding the radius of convergence helps identify where this series converges.
• The center \( z_0 \) plays a crucial role, and the series will converge within a certain distance from this point.

Complex power series are used in various fields, such as quantum mechanics and electrical engineering, where complex numbers frequently appear.

Circle of Convergence

The **circle of convergence** is a circle in the complex plane centered at the point \( z_0 \) where the complex power series is centered. The radius of this circle is the radius of convergence, and it defines the boundary within which the series converges.

• Inside this circle, the series converges absolutely and uniformly.
• On the circle's boundary, convergence behavior can vary and needs individual testing.
• Outside this circle, the series diverges.

Understanding the circle of convergence is fundamental when dealing with complex functions, as it informs you where the function, represented by a power series, can be applied safely.

Limit Superior

The **limit superior**, often denoted as \( \limsup \), is a concept in calculus used to describe the behavior of sequences. For finding the radius of convergence of a complex power series, this concept is crucial.

• The formula \( \frac{1}{R} = \limsup_{k \to \infty} \sqrt[k]{|a_k|} \) helps calculate the radius of convergence \( R \).
• The limit superior considers the greatest limit point of a sequence, providing a sort of upper boundary for the sequence's behavior as \( k \) grows very large.
• In our solution, \( \limsup \) was used to determine the limit of \( \sqrt[k]{|a_k|} \) as \( k \) approaches infinity, leading us to isolate \( R \).

Limit superior can often help in complex analysis not just for convergence, but also for evaluating the behavior of various sequences and series.

Complex Analysis

**Complex analysis** is a branch of mathematics that studies functions of complex numbers. It extends the idea of calculus to the complex number plane, providing tools to analyze complex power series and other complex functions.

• Complex analysis introduces concepts such as holomorphic functions, conformal mappings, and residues.
• It allows us to explore properties and applications of series, such as those in our example, by involving the entire plane of complex numbers.
• This field is pivotal in solving integrals and differential equations where ordinary real methods fall short.

Complex analysis is not only foundational in pure mathematics but also has applications in physics, engineering, and fields that involve complex systems.