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Radius of Convergence

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Radius of Convergence

Study Guide

Key Definition

The radius of convergence is the radius of the disk within which a power series $\sum_{n=0}^{\infty} c_n (z-a)^n$ converges.

Important Notes

The radius of convergence is a non-negative real number or can be infinite ($\infty$).
If $|x-a| \lt r$, the series converges.
If $|x-a| \gt r$, the series diverges.
Convergence is absolute and uniform on any closed subinterval within the interval of convergence.
The center of convergence, $a$, is crucial in determining the radius.

Mathematical Notation

$|x-a| \lt r$ indicates convergence within the radius.

$\sum$ denotes summation.

$\infty$ represents infinity.

Remember to use proper notation when solving problems

Why It Works

The concept of the radius of convergence ensures that a power series converges only for values within a specific distance from the center. This is crucial for the analysis of functions represented by power series.

Remember

The radius of convergence is determined by the formula $R = \frac{1}{\limsup_{n \to \infty} \sqrt[n]{|c_n|}}$.

Quick Reference

Convergence Condition:$|x-a| \lt r$

Understanding Radius of Convergence

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Imagine a circle around the center $a$. The series converges for any $x$ within the circle's radius $r$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

For the series $\sum_{n=0}^{\infty} c_n (x-a)^n$, if $|x-a| = 2$ and the radius of convergence $r = 3$, does the series converge?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are tuning a radio and need to know the range of frequencies it can pick up. This is similar to finding the radius of convergence for a power series.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given the series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$, determine the radius of convergence.

Challenge Quiz

Single Choice Quiz

Advanced

For the power series $\sum_{n=0}^{\infty} \frac{(x-2)^n}{2^n}$, what is the radius of convergence?

Recap

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Review key concepts and takeaways