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, which is 1/2 of the interval of convergence.

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

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.

Also see: D’Alembert’s Ratio Test

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:

$$R = \inf |z – \psi|: z \in \mathbb{C}, \sum_{n= 0}^\infty a_n {(z – \psi)}^n \mathbf{ is divergent}$$

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:

  1. At this calculator page, enter the function and range in their respective input fields.
  2. Click the Calculate button to get the output.
  3. 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.

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