Radius of convergence of sum ((z-2i)^n)/(n^n) as n approaches infinity from 0

The given series is:

\[
\sum_{n=0}^{\infty} \frac{(z - 2i)^n}{n^n}
\]

To find the radius of convergence of this power series, we can apply the **Root Test** (Cauchy-Hadamard theorem). The radius of convergence \( R \) is given by:

\[
\frac{1}{R} = \limsup_{n \to \infty} \sqrt[n]{|a_n|}
\]

where \( a_n = \frac{(z - 2i)^n}{n^n} \).

### Step 1: Find the general term

The general term is:

\[
a_n = \frac{(z - 2i)^n}{n^n}
\]

We can express this as:

\[
a_n = \left( \frac{z - 2i}{n} \right)^n
\]

### Step 2: Apply the Root Test

We now apply the root test to this series. Taking the \( n \)-th root of \( a_n \):

\[
\sqrt[n]{|a_n|} = \sqrt[n]{\left| \frac{(z - 2i)^n}{n^n} \right|}
\]

Simplifying:

\[
\sqrt[n]{|a_n|} = \left| \frac{z - 2i}{n} \right|
\]

This simplifies further to:

\[
\sqrt[n]{|a_n|} = \frac{|z - 2i|}{n}
\]

### Step 3: Find the limit

Taking the limit as \( n \to \infty \):

\[
\lim_{n \to \infty} \frac{|z - 2i|}{n} = 0
\]

Since this limit is 0, the series converges for all \( z \). Therefore, the radius of convergence \( R \) is infinite.

### Final Answer:

The radius of convergence is \( \boxed{R = \infty} \).

Would you like more details on how the Root Test works, or any additional explanations?

Here are 5 related questions you could explore:
1. How does the Root Test compare to the Ratio Test for finding the radius of convergence?
2. What is the significance of an infinite radius of convergence for a power series?
3. How would the radius of convergence change if the general term were \( \frac{(z - 2i)^n}{n!} \)?
4. What are some common series with finite radii of convergence?
5. How do singularities of a function affect the radius of convergence of its series expansion?

**Tip:** When applying the Root Test, the behavior of \( a_n \) as \( n \to \infty \) is key to determining convergence.

Discover how to find the radius of convergence for the power series ((z-2i)^n)/(n^n) using the Root Test (Cauchy-Hadamard Theorem). This problem explores key concepts in complex analysis, including the general term of the series and an infinite radius of convergence. Ideal for undergraduate students in calculus or complex analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation