Radius of Convergence and Interval of Convergence Calculator
In recent times, online calculators have gained a lot of popularity. The radius of convergence calculator, also known as the interval of convergence calculator, is a free online resource that gives you the convergence point for a given series.
The radius of convergence is a concept in calculus, real & complex analysis, related to the interval of convergence as described below.
Table of Contents
Radius of Convergence
The radius of convergence of a series is a number $R$ for which the power series, $\sum\limits_{n = 0}^\infty {{c_n}{{\left( {x - a} \right)}^n}}$ will converge for $|x−a|<R$ ; and will diverge for $|x−a|>R$.
Note that the series may or may not converge if $|x−a|=R$. What happens at these points will not change the radius of convergence
Interval of Convergence
The interval of convergence of a series, as the name suggests, is the set of values (an interval) for which the series, mainly a power series, is converging.
In the above example, the interval of convergence will be $(a-R, a+R)$.
See D'Alembert's ratio test for more material on this.
The Radius of Convergence Calculator
This calculator is also an Interval of convergence calculator as it offers complete solutions on what the radius and interval of a convergence series will be.
Using this form you can calculate the radius of convergence. Say, if you put n(x-3)^n/2^n, where n tends from 1 to infinity; you'd literally mean $\sum\limits_{n = 1}^\infty {\dfrac{n(x-3)^n}{2^n}}$
The calculator will pop up with the following answer.
$\sum\limits_{n = 1}^\infty {\dfrac{n(x-3)^n}{2^n}}$ converges when $|x - 3|<2$
So your radius of convergence here is 2, and the interval of convergence will be (3-2,3+2) or (1,5).
You can change the values and calculate using the Radius of Convergence Calculator this way.
More on 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.
Radius of Convergence in Real Set R
Let $\psi \in \mathbb{R}$ be a real number.
Let $\displaystyle S(x) = \sum_{n \mathop = 0}^\infty a_n {(x - \psi)}^n$ be a power series about $\psi$.
Let $I$ be the interval of convergence of $S(x)$.
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 $S(x)$.
Radius of Convergence in Complex Set C
Let $\psi \in \mathbb{C}$ be a complex number.
For $z \in \mathbb{C}$, let:
$\displaystyle f(z) = \sum_{n= 0}^\infty a_n {(z - \psi)}^n$ be a power series about $\psi$.
The radius of convergence is the extended real number $R \in \overline{\mathbb{R}}$ defined by:
$$R = \inf |z - \psi|: z \in \mathbb{C}, \sum_{n= 0}^\infty a_n {(z - \psi)}^n \mathbf{ is divergent}$$
Important Power Series and Their Interval of Convergence
| Function/Power Series | Interval of Convergence |
|---|---|
| $\sin(x)=\sum^\infty_{n=0}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}$ | $\mathbb{R}$ |
| $\cos(x)=\sum^\infty_{n=0}\frac{(-1)^{n}x^{2n}}{(2n)!}$ | $\mathbb{R}$ |
| $\sinh(x)=\sum^\infty_{n=0}\frac{x^{2n+1}}{(2n+1)!}$ | $\mathbb{R}$ |
| $\cosh(x)=\sum^\infty_{n=0}\frac{x^{2n}}{(2n)!}$ | $\mathbb{R}$ |
| $\tan^{-1}(x)=\sum^\infty_{n=0}\frac{(-1)^{n}x^{2n+1}}{2n+1}$ | $|x|<1$ |
| $e^x=\sum^\infty_{n=0}\frac{x^n}{n!}$ | $\mathbb{R}$ |
| $\frac{1}{1-x}=\sum^\infty_{n=0}x^n$ | $|x|<1$ |
| $\frac{1}{1+x}=\sum^\infty_{n=0}(-1)^{n}x^n$ | $|x|<1$ |
| $\ln(x)=\sum^\infty_{n=1}\frac{(-1)^{n}(x-1)^n}{n}$ | $|1-x|<1$ |
| $\ln(1+x)=\sum^\infty_{n=1}\frac{(-1)^{n+1}x^n}{n}$ | $|x|<1$ |
| $\ln(1-x)=-\sum^\infty_{n=1}\frac{x^n}{n}$ | $|x|<1$ |
| $\ln(\frac{1+x}{1-x})=2\sum^\infty_{n=0}\frac{x^{2n+1}}{2n+1}$ | $|x^2|<1$ |
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.
It doesn't only give you the radius of convergence of a series but also the interval of the convergence of the series. In addition to that, my calculator here draws a plot representing the series so that you can understand how the series looks and where the points of interval are located.
The radius of convergence calculator is a useful tool and is a highly recommended resource for teachers, students, and even professional mathematicians.
Important Notes
- The radius of convergence is always a positive real number.
- The interval of convergence can be open, closed, or half-open, depending on the power series' behavior at the interval's endpoints.
- The radius of convergence and interval of convergence are related but not the same. The interval of convergence is the set of real numbers for which the power series converges, while the radius of convergence gives the boundary beyond which the power series diverges.
- In other words, the interval of convergence can be larger or smaller than the circle with radius R, depending on the power series' behavior at the interval's endpoints.
