Problem 21 - Advanced Engineering Mathematics

Question

Find the circle and radius of convergence of the given power series. \(\sum_{k=0}^{\infty} \frac{1}{(1-2 i)^{k+1}}(z-2 i)^{k}\)

Step-by-Step Solution

Answer

The circle of convergence is centered at \(2i\) with radius \(\sqrt{5}\).

Step 1: Identify the general form of the power series

The power series given is \(\sum_{k=0}^{\infty} \frac{1}{(1-2i)^{k+1}}(z-2i)^{k}\). This is essentially a geometric series of the form \(\sum_{k=0}^{\infty} a_k (z - c)^k\), where \(a_k = \frac{1}{(1-2i)^{k+1}}\) and \(c = 2i\). In a geometric series, the common ratio \(r\) for each term is \(\frac{a_{k+1}}{a_k} = (z-c)\).

Step 2: Determine the radius of convergence using the formula for geometric series

The radius of convergence \(R\) of a geometric series \(\sum_{k=0}^{\infty} a_k (z - c)^k\) is determined by the formula \(|r| < 1\), where \(r = \frac{1}{1-2i}\) since \(a_{k+1} = \frac{1}{(1-2i)^{k+2}}\) and \(a_k = \frac{1}{(1-2i)^{k+1}}\). The circle of convergence is centered at \(c = 2i\).

Step 3: Calculate the modulus of the common ratio

To find the modulus (absolute value) of the common ratio \(r = \frac{1}{1-2i}\), we first need to find the modulus of \(1-2i\). The modulus is given by \(|1-2i| = \sqrt{1^2 + (-2)^2} = \sqrt{5}\). Thus, \(|r| = \left|\frac{1}{1-2i}\right| = \frac{1}{|1-2i|} = \frac{1}{\sqrt{5}}\).

Step 4: Establish the radius of convergence

From the modulus calculation, \(|r| = \frac{1}{\sqrt{5}} < 1\), so the radius of convergence \(R\) is \(\sqrt{5}\). This means the series converges when \(|z - 2i| < \sqrt{5}\).

Step 5: Conclude the circle and radius of convergence

The circle of convergence is centered at \(2i\) (the point \(c\)) and has a radius \(R = \sqrt{5}\). Therefore, the power series converges within the circle \(|z - 2i| < \sqrt{5}\).

Key Concepts

Radius of Convergence

When dealing with power series, one of the key concepts is the **radius of convergence**. This is a measure that defines where exactly a power series will converge or behave nicely. More technically, for a power series centered around some point, say \(c\), the series will converge for values of \(z\) that lie within a "radius" around this point. In simpler terms, if you imagine a circle around the center \(c\), any \(z\) within this circle will make the series converge.

To find this radius, we use formulas specific to the kind of series we're dealing with. For the geometric series, there's a handy formula: the series converges when the absolute value of the common ratio \(r\) is less than one, i.e., \(|r| < 1\).

Calculating \(|r|\) often involves finding the modulus of complex numbers (think of them like distances in the complex plane). In the case of the given problem, the radius \(R = \sqrt{5}\) tells us the series behaves properly as long as \(|z - 2i| < \sqrt{5}\).

Power Series

A **power series** is a special form of infinite series. Think of it as a long polynomial where each term is a power of \(z\). It's generally written as \(\sum_{k=0}^{\infty} a_k (z-c)^k\) where each \(a_k\) is a coefficient and \(c\) is the center around which the series expands.

This form is flexible and useful in approximating functions that are otherwise hard to calculate. The major aspect of power series is knowing where they converge, which brings us back to the radius of convergence. **Convergence** means that if you add up infinitely many terms of the series, it will settle on some finite number. This is crucial in calculus and complex analysis because it helps to understand the behavior of complex functions.

In practical terms, a power series can be used to express functions like sin, cos, or even more complex ones, and the radius of convergence tells us exactly where these expressions are correct.

Geometric Series

The **geometric series** is one of the simplest forms of series and a specific type of power series. Each term in a geometric series is a constant multiple of the previous one. Think of it like stacking blocks, where each block is a fraction of the size of the last. Mathematically, it looks like \(\sum_{k=0}^{\infty} ar^k\), where \(a\) is the first term and \(r\) is the common ratio.

The beauty of geometric series is in their simplicity and the neat formulas we can use to sum them up. Particularly, a geometric series converges (adds up to a finite value) if and only if the absolute value of the common ratio \(|r|\) is less than one. In that case, the sum equals \(\frac{a}{1-r}\).

In the exercise we're considering, identifying the power series as ultimately a geometric series helped us easily find the radius of convergence. The common ratio was to ensure convergence within the defined circle of convergence. Thus, recognizing a geometric series can greatly simplify complex problems in calculus and beyond.