Problem 28 - A First Course in Complex Analysis with Applications

Question

Find the circle and radius of convergence of the given power series. $$ \sum_{k=0}^{\infty} \frac{k !}{(2 k)^{k}} z^{3 k} $$

Step-by-Step Solution

Answer

The circle of convergence is \( |z| < (2e)^{1/3} \).

Step 1: Identify the general form

First, observe that the power series is given as \( \sum_{k=0}^{\infty} \frac{k!}{(2k)^{k}} z^{3k} \). In this format, the general term of the series is \( a_k z^{3k} \) where \( a_k = \frac{k!}{(2k)^k} \).

Step 2: Use the Ratio Test

To find the radius of convergence, we apply the Ratio Test. We calculate \( \lim_{k \to \infty} \left| \frac{a_{k+1} z^{3(k+1)}}{a_k z^{3k}} \right| \). This simplifies to \[ \lim_{k \to \infty} \left| \frac{(k+1)!}{(2k+2)^{k+1}} \cdot \frac{(2k)^k}{k!} \cdot z^3 \right|. \]

Step 3: Simplify the ratio

Simplify the ratio: \[ \lim_{k \to \infty} \left| \frac{(k+1)k!}{k!} \cdot \left(\frac{2k}{2(k+1)}\right)^k \cdot \frac{1}{2(k+1)} \cdot z^3 \right|. \] Further simplification yields \[ \frac{k+1}{2(k+1)} \cdot \left( \frac{k}{k+1} \right)^k \cdot |z|^3. \]

Step 4: Evaluate the limit

As \( k \) approaches infinity, notice that \( \left( \frac{k}{k+1} \right)^k \to \frac{1}{e} \) and \( \frac{k+1}{2(k+1)} \to \frac{1}{2} \). Thus, the limit becomes \( \frac{|z|^3}{2e} \).

Step 5: Determine the radius of convergence

Set \( \frac{|z|^3}{2e} < 1 \) for convergence, leading to \( |z|^3 < 2e \). Taking the cube root gives \( |z| < (2e)^{1/3} \). Thus, the radius of convergence is \( (2e)^{1/3} \).

Step 6: Identify the circle of convergence

The circle of convergence in the complex plane is defined by \( |z| < (2e)^{1/3} \). This circle is centered at the origin with radius \( (2e)^{1/3} \).

Key Concepts

Power Series

A power series is a type of infinite series that looks like a polynomial but has infinitely many terms. Its general form is given by \( \sum_{k=0}^{\infty} a_k z^k \), where each term is composed of a coefficient \( a_k \) and a variable raised to increasing powers. The variable \( z \) can be a real or complex number.

• The coefficients \( a_k \) can be calculated based on the formula related to the specific problem or pattern of the series.
• Power series are widely used because they allow the expression of complex functions in a more manageable form.

The series becomes a helpful tool because the behavior of the function can be studied by analyzing the power series instead, which is often easier to manipulate and explore.

Radius of Convergence

The radius of convergence of a power series is the distance from the center of the series within which the series converges. To find it, we consider how far the series terms decrease as more are added.

• Mathematically, if the power series \( \sum a_k z^k \) converges for \( |z| < R \) and diverges for \( |z| > R \), then \( R \) is called the radius of convergence.
• The radius of convergence can be zero, finite, or infinite. If it is infinite, the series converges everywhere in the complex plane.

For the given series, the radius was determined using the Ratio Test, leading to \( R = (2e)^{1/3} \). Understanding the concept of the radius is essential because it tells us where the series provides meaningful and convergent results.

Ratio Test

The Ratio Test is a mathematical tool used to determine whether an infinite series converges or diverges. It is commonly applied to power series to find their radius of convergence.

• The test involves taking the limit of the absolute value of the ratio of consecutive terms in the series: \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \).
• If this limit is less than one, the series converges absolutely. If greater than one, it diverges.

Applying the Ratio Test to our problem, we worked with the terms \( \frac{k!}{(2k)^k} \). Through simplification, we discovered the radius of convergence as previously mentioned. The Ratio Test is vital as it elegantly narrows down the range in which our series is valid for convergence.

Circle of Convergence

The circle of convergence for a power series is depicted in the complex plane as a circle centered at the origin with radius equal to the radius of convergence. This circle shows where the series behaves well and is applicable.

• Geometrically, it is the boundary within which the series will converge for any point \( z \) inside the circle.
• If \( |z| < R \), within the circle, the series converges. If \( |z| > R \), outside the circle, it diverges.

For our series, the circle of convergence is defined by the inequality \( |z| < (2e)^{1/3} \), meaning any point within this circle will ensure the series behaves nicely. Understanding both the radius and circle of convergence gives us a complete picture of how and where we can use our power series effectively.