Question

Find the disk of convergence for each of the following complex power series. $$\sum_{n=1}^{\infty} \frac{(z-i)^{n}}{n}$$

Short answer

The disk of convergence is centered at i with a radius of 1.

Step-by-step solution

Identify the given complex power series

The given complex power series is \[ \sum\_{n=1}^{\infty} \frac{(z-i)^{n}}{n} \].

Define the general term of the series

Express the general term of the series. In this case, the general term is \ \[ a_n \ = \ \frac{(z-i)^{n}}{n} \ \].

Apply the ratio test for convergence

The ratio test states that the series \ \[ \sum_{n=1}^{\infty} a_n \ \] converges if \ \[ \lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \ \]. Calculate the ratio \ \[ \left| \frac{a_{n+1}}{a_n} \right| \ = \ \frac{\left|(z-i)^{n+1}\right|}{(n+1)} \cdot \frac{n}{\left|(z-i)^n\right|} \ = \ \frac{\left|(z-i)\right|}{1+\frac{1}{n}} \ \].

Simplify the limit

Simplify the expression as \[ n \rightarrow \infty \]. The ratio becomes \ \[ \lim_{n \rightarrow \infty} \frac{\left| z-i \right|}{1+\frac{1}{n}} = \left| z-i \right| \].

Set up the inequality for convergence

For the series to converge, we need \ \[ \left| z-i \right| < 1 \ \]. This inequality represents the set of points within a distance of 1 from \ i \ in the complex plane.

Determine the disk of convergence

The disk of convergence is centered at \ i \ with a radius of 1. Therefore, the power series converges for all \ z \ such that \ \[ \left| z-i \right| < 1 \ \].

Key concepts

disk of convergence

To understand the concept of the **disk of convergence**, let's start with our given series that the area in the complex plane where the series converges is called the disk of convergence. Visualize it as a circle centered at a point (in this instance, i ). The radius is the distance from the center to the circumference where our series converges. For our problem, we found that the radius of the disk is 1 , from: \[ \left|z - i\right| < 1 \] Our disk of convergence therefore includes all points z within a distance of 1 from i .

ratio test

The “complex**” “essentially can” “sum up in” “a” two-dimensional plane where complex numbers are represented. Complex **numbers** have the form: \ \ \ \ \[z = a + bi” \] Different “parts” “plus a real----: \ term.) ”