Problem 28 - Advanced Engineering Mathematics

Question

From \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|(-1)^{n}\left(\frac{1+2 i}{2}\right)^{n}\right|}=\lim _{n \rightarrow \infty}\left|\frac{1+2 i}{2}\right|=\frac{\sqrt{5}}{2}\) we see that the radius of convergence is \(R=2 / \sqrt{5}\). The circle of convergence is \(|z+2 i|=2 / \sqrt{5}\).

Step-by-Step Solution

Answer

The radius of convergence is \(\frac{2}{\sqrt{5}}\); the circle of convergence is \(|z + 2i| = \frac{2}{\sqrt{5}}\).

Step 1: Interpret the given expression

The problem deals with finding the radius of convergence from the expression \(\lim _{n \rightarrow \infty} \sqrt[n]{\left|(-1)^{n}\left(\frac{1+2i}{2}\right)^{n}\right|}\). This expression determines the radius of convergence of a power series around a specific point in the complex plane.

Step 2: Simplify the expression

First, recognize that \((-1)^n\) is the factor that simply oscillates between -1 and 1, which does not affect the magnitude of the terms. The expression simplifies by focusing on \(\left(\frac{1+2i}{2}\right)^{n}\), more specifically its absolute value: \(\left|\frac{1+2i}{2}\right|^n\).

Step 3: Calculate the absolute value

To find \(\left|\frac{1+2i}{2}\right|\), use the formula for the magnitude of a complex number: \(\left|a + bi\right| = \sqrt{a^2 + b^2}\). Here, \(a = 0.5\) and \(b = 1\), so the magnitude is \(\sqrt{0.5^2 + 1^2} = \sqrt{0.25 + 1} = \sqrt{1.25}\). Hence, \(\left|\frac{1+2i}{2}\right| = \frac{\sqrt{5}}{2}\).

Step 4: Apply the Root Test

According to the root test for convergence, the limiting expression \(\lim _{n \rightarrow \infty} \sqrt[n]{|a_n|}\) gives the radius of convergence \(R\) as \(R = \frac{1}{\lim _{n \rightarrow \infty} \sqrt[n]{|a_n|}}\). With our calculated \(\left|\frac{1+2i}{2}\right| = \frac{\sqrt{5}}{2}\), the radius of convergence is \(R = \frac{1}{\frac{\sqrt{5}}{2}} = \frac{2}{\sqrt{5}}\).

Step 5: Determine the circle of convergence

The circle of convergence in the complex plane is given by the circle centered at the series' expansion point (which appears to be \(-2i\) from the context) with radius \(R = \frac{2}{\sqrt{5}}\). Therefore, the circle is defined by \(|z + 2i| = \frac{2}{\sqrt{5}}\).

Key Concepts

Radius of Convergence

In complex analysis, the radius of convergence is a fundamental concept when dealing with power series. It determines the "distance" in the complex plane within which a power series converges.
For any power series of the form \( \sum_{n=0}^{\infty} a_n z^n,\) the radius of convergence \( R \) is a non-negative real number that marks the boundary.

• If \( |z| < R \), the series converges absolutely.
• If \( |z| > R \), the series diverges.
• If \( |z| = R \), further analysis, such as the Root or Ratio Test, is required to check convergence.

Calculating \( R \) involves mathematical techniques like the Root Test. Here, the key is understanding how \( R \) demonstrates the "reach" of a series within the complex plane, which in turn dictates where a series represents a valid, convergent function.

Root Test

The Root Test is a valuable tool we use to determine the radius of convergence for a power series. It's particularly useful when the terms of the series are raised to increasing powers. Let's break it down step-by-step.
Suppose you have a power series \( \sum a_n z^n \). The Root Test calculates:\[L = \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|}\]

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

To find the radius of convergence \( R \), apply the formula:\[R = \frac{1}{L}\]In our example, the absolute value expression \( \left| \frac{1+2i}{2} \right| \) simplifies to \( \frac{\sqrt{5}}{2} \). Hence, \( L = \frac{\sqrt{5}}{2} \) and \( R = \frac{2}{\sqrt{5}} \). The radius effectively encapsulates the extent of convergence for the given series in the complex plane.

Complex Numbers

Complex numbers extend the realm of standard numbers by including imaginary portions. A general complex number is formulated as \( a + bi \), where:

• \( a \) represents the real part.
• \( b \) represents the imaginary part.
• \( i \) is the imaginary unit, defined by the property \( i^2 = -1 \).

In our example, we're dealing with the complex number \( \frac{1+2i}{2} \). To find its magnitude, which is crucial for convergence tests, apply the magnitude formula: \( \sqrt{a^2 + b^2} \). For \( a = 0.5 \) and \( b = 1 \), the calculation becomes \( \sqrt{0.5^2 + 1^2} = \sqrt{1.25} = \frac{\sqrt{5}}{2} \).
Complex numbers allow power series to expand into different planes, making convergence analysis more expansive than real-number-only scenarios.

Power Series

A power series is essentially an infinite polynomial, serving as a cornerstone in both real and complex analysis. Typically, it looks like \( \sum_{n=0}^{\infty} a_n (z-c)^n \), where:

• \( a_n \) is the series coefficient at each iteration.
• \( z \) represents the complex variable.
• \( c \) is the center about which the series is expanded.

The attractiveness of a power series comes from its descriptive power. It can represent functions in a way that makes differentiation and integration straightforward.
In the realm of complex analysis, every power series has a circle of convergence. This is the area within which the series converges to a well-defined value. In our context, the circle is centered at \( -2i \), and its radius is determined by the previously calculated \( R = \frac{2}{\sqrt{5}} \). This means the series converges for all \( z \) such that \( |z+2i| < \frac{2}{\sqrt{5}} \). Understanding power series enables us to map complex behaviors into calculable scopes, unlocking deeper insights into calculus and analysis.