# Radius of Convergence Calculator

Traditional mathematicians generally frown upon the usage of calculators. However, calculators can help students and professionals in solving complicated problems efficiently and quickly and efficiently. They help us save time and make mathematics more interesting.

In recent times, online calculators have gained a lot of popularity as well. Radius of convergence calculator is such a free online resource which gives you the convergence point for a given series.

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## What Is The Radius of Convergence?

The power series converges at the center of its convergence at a particular interval. The radius of convergence is the distance from the center of convergence to the other end of the interval.

For a particular power series, it is calculated using the ratio test. It is considered the best test to calculate the convergence that instructs to calculate the limit. When the limit is less than 1, this test can be used to accurately predict the convergence point.

Also see: D’Alembert’s Ratio Test

### Radius of Convergence in Real Set R

Let $\psi \in \R$ be a real number.

Let $\displaystyle \map S x = \sum_{n \mathop = 0}^\infty a_n \paren {x – \psi}^n$ be a power series about $\psi$.

Let $I$ be the interval of convergence of $\map S x$.

Let the endpoints of $I$ be $\psi – R$ and $\psi + R$.

(This follows from the fact that $\psi$ is the midpoint of $I$.)

Then $R$ is called the **radius of convergence** of $\map S x$.

### Radius of Convergence in Complex Set C

Let $\psi \in \C$ be a complex number.

For $z \in \C$, let:

$\displaystyle \map f z = \sum_{n \mathop = 0}^\infty a_n \paren {z – \psi}^n$

be a power series about $\psi$.

The radius of convergence is the extended real number $R \in \overline \R$ defined by:

$R = \displaystyle \inf \set {\cmod {z – \psi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z – \psi}^n \text{ is divergent} }$

## How to Use the Radius of Convergence Calculator

This radius of convergence calculator has a simple and user-friendly interface. You can use it by following the steps given below:

- At this calculator page, enter the function and range in their respective input fields.
- Click the Calculate button to get the output.
- The convergence point for the given series will be shown in a new window which opens up automatically.

**What Are The Advantages Of Using The Radius of Convergence Calculator?**

Our radius of convergence calculator is free to use and can be used by anyone with a working internet connection. It is fast, accurate, and saves you a lot of time.

The radius of convergence calculator is a useful tool and is a highly recommended resource for teachers, students, and even professional mathematicians.