Problem 27 - Advanced Engineering Mathematics

Question

Find the circle and radius of convergence of the given power series. \(\sum_{k=0}^{\infty} \frac{(z-4-3 i)^{k}}{5^{2 k}}\)

Step-by-Step Solution

Answer

The circle of convergence is \(|z - (4 + 3i)| < 25\), with a radius of 25.

Step 1: Identify the Power Series Format

Given the power series \( \sum_{k=0}^{\infty} \frac{(z-4-3i)^{k}}{5^{2 k}} \), identify if it fits the standard power series format \( \sum_{k=0}^{\infty} c_k (z-a)^{k} \). Here, \( c_k = \frac{1}{5^{2k}} \) and \( a = 4 + 3i \).

Step 2: Apply the Ratio Test to Determine Convergence

For a power series \( \sum_{k=0}^{\infty} c_k (z-a)^{k} \), the radius of convergence \( R \) is given by \( R = \frac{1}{\lim_{k \to \infty} \left| \frac{c_{k+1}}{c_k} \right|} \). Here, \( c_k = \frac{1}{5^{2k}} \) and \( c_{k+1} = \frac{1}{5^{2(k+1)}} \). Hence, \( \frac{c_{k+1}}{c_k} = \frac{1}{5^2} \).

Step 3: Calculate the Radius of Convergence

From the formula, find \( R = \frac{1}{\left| \frac{1}{5^2} \right|} = 25 \). Thus, the radius of convergence is 25.

Step 4: Determine the Circle of Convergence

The power series converges inside the circle centered at \( a = 4 + 3i \) with radius 25. Therefore, the circle of convergence is \( |z - (4 + 3i)| < 25 \).

Key Concepts

Power Series

A power series is like an infinite summation of terms with powers of a variable. It looks like this:

• \(\sum_{k=0}^{\infty} c_k (z-a)^{k}\)

where \(c_k\) represents the coefficients and \(a\) is the center of the series. Here, the series is formed by raising expressions to powers and summing them.

Think of a power series as similar to a polynomial, but with infinitely many terms. The term

• \(z-a\)

represents a shift of the variable \(z\) from some point \(a\) on the complex plane. Each term of the power series increases in power, which means each term involves higher powers of \(z-a\).

In the exercise given, the power series was

• \(\sum_{k=0}^{\infty} \frac{(z-4-3 i)^{k}}{5^{2 k}}\).

The center here is \(a = 4 + 3i\), and the coefficient for each term is \(c_k = \frac{1}{5^{2k}}\).

Radius of Convergence

The radius of convergence tells us how far away from the center \(a\) the power series converges. This means within this radius, the series sums to a finite value. Beyond this radius, the series may not behave nicely (it might not sum to a finite value).

The radius of convergence ensures that we know the boundary of the region where our power series behaves predictably. Consider it as a key to unlocking the area on a complex plane where our series actually makes sense.

• Inside the circle with this radius: The series converges
• Outside this circle: The series may diverge

In our exercise, the power series centered at \(4 + 3i\) was found to have a radius of convergence of 25. Thus, the series converges within this circle around \(4 + 3i\). Beyond 25 units away, we can't be certain of convergence.

Ratio Test

The ratio test is a method to determine the radius of convergence of a power series. It's all about looking at the behavior of the coefficients as the series progresses.

Here's how you apply it:

• Take the limit of the ratio of sequential coefficients: \( \lim_{k \to \infty} \left| \frac{c_{k+1}}{c_k} \right| \).
• The radius of convergence \(R\) is \( \frac{1}{\text{{result of the limit}}} \).

The step-by-step solution applies the ratio test to find the radius of convergence. For our given series, the sequence of coefficients is \(c_k = \frac{1}{5^{2k}}\). By calculating

\( \frac{c_{k+1}}{c_k} = \frac{1}{5^2} \), and thus finding \( \lim_{k \to \infty} \left| \frac{c_{k+1}}{c_k} \right| = \frac{1}{25} \), we determine the radius is 25.
The ratio test is efficient and reliable, providing a clear path to understanding convergence.