Problem 23 - Advanced Engineering Mathematics

Question

Find the circle and radius of convergence of the given power series. \(\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k 2^{k}}(z-1-i)^{k}\)

Step-by-Step Solution

Answer

The circle of convergence is centered at \(1+i\) with a radius of 2.

Step 1: Identify the Center of the Circle of Convergence

The given power series is \[ \sum_{k=1}^{\infty} \frac{(-1)^{k}}{k 2^{k}}(z-1-i)^{k}.\] This series is centered at the point \(1+i\). This means that the circle of convergence is centered at \(z = 1 + i\).

Step 2: Apply the Ratio Test to Find the Radius of Convergence

To find the radius of convergence, we use the Ratio Test. The Ratio Test involves finding the limit \[ L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|,\] where \(a_k = \frac{(-1)^k}{k 2^k}.\)

Step 3: Compute the Ratio of Consecutive Terms

Calculate the ratio \[ \left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{\frac{(-1)^{k+1}}{(k+1) 2^{k+1}}}{\frac{(-1)^{k}}{k 2^{k}}} \right| = \frac{k}{k+1} \cdot \frac{1}{2},\] which simplifies to \( \frac{k}{k+1} \cdot \frac{1}{2}.\)

Step 4: Determine the Limit for the Ratio Test

Find the limit as \(k\) approaches infinity:\[ L = \lim_{k \to \infty} \frac{k}{k+1} \cdot \frac{1}{2} = \frac{1}{2}, \] indicating convergence when \( |z - (1+i)| < 2.\)

Step 5: Conclude the Radius of Convergence

Since the series converges if \[ \left| z - (1+i) \right| < 2,\] this means the radius of convergence is 2.

Key Concepts

Power Series

Power series are like polynomials that stretch towards infinity. They are often written in the form: \[ \sum_{k=0}^{ ext{∞}} a_k (z - c)^k,\] where \(a_k\) are the coefficients, \(z\) is a complex variable, and \(c\) is the center of the series.
Power series can be used to express functions as an infinite sum of terms calculated from the values and derivatives of a function at a single point. The region where the series converges, or sums to a certain value for the complex numbers, is critical.
For example, in the exercise, the power series is centered at \(z = 1 + i\), meaning each term is a power of \((z - (1+i))\). This center is like the "home base" of the series where all the calculations start. Understanding where a power series sums up, and thus "works," is fundamental in both real and complex analysis.

Ratio Test

The Ratio Test is a valuable tool for determining the convergence of a power series. It checks whether the terms in the series keep getting smaller fast enough. Here's how it works:
The test involves computing the limit:\[L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|,\]where \(a_k\) is the series term. If \(L < 1\), the series converges absolutely. This means the endless sum approaches a finite number, a crucial understanding when analyzing series.
In the exercise, the Ratio Test helped to find the radius of convergence of the series. By simplifying the ratio of consecutive terms and finding the limit \(L\), we concluded that the series converges if \(|z - (1+i)| < 2\). Thus, the radius of convergence is 2, determining how far from its center the power series still adds up to a meaningful result.

Complex Analysis

Complex analysis explores functions of complex numbers and dives deeper into the heart of mathematical functions.
Unlike usual numbers, complex numbers carry both a real and an imaginary part, often expressed as \(z = x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit \(\sqrt{-1}\).
One significant focus in complex analysis is understanding how power series behave in the complex plane. This involves analyzing how these series converge depending on their distance from a center point, leading us to concepts like the circle of convergence.
In the exercise, knowing how to handle the imaginary component \((1+i)\) was vital in finding the center of the circle of convergence. Complex analysis allows mathematic functions to be extended and interpreted in spaces where they are less intuitive than in the real numbers and where beautiful mathematical properties often emerge, such as analytic continuation and residue theorem.