Radius of Convergence Calculator

Radius of Convergence Calculator

Enter the function, select a variable, and click 'Calculate' to determine the radius of convergence of a power series.

Radius of Convergence Calculator

This Radius of convergence calculator is specially designed to calculate the radius of convergence of a given power series. It is the best tool for identifying that where the series converges. For user convenience, the convergence radius calculator shows the step-by-step solution.

What is Radius of Convergence?

“The radius of convergence is the maximal radius of a disk centered at a series within which a series converges”

It is centered at a specific point in the non-negative real number denoted by R such that:

If |x - a| < R, the series converges
If |x - a| > R, the series diverges

How do you Find the Radius of Convergence?

The root test and ratio tests are used to find the radius of convergence so look at these.

Ratio Test:

It is one of the tests that is used to find the convergence, divergence, radius of convergence, and interval of convergence.

$$ L= \lim_{n \to \infty} \frac{a_{n+1}} {a_n} $$

Root Test:

The root test is the test for a series when there raised to the nth power without any factorial expression. Likewise to the ratio test, the convergence depends on the value of the limit.

$$ L = \lim_{n\to\infty}\left|a_n^{\frac{1}{n}}\right| $$

Both given tests also help analyze the interval of convergence, which specifies the range of values for which the series converges. If you're looking for a quick way to compute these values, you can use an interval of convergence calculator to automate and verify your results. Look at the example that implements these tests in calculations.

Example:

Find the radius of convergence, R, of the series below.

$$ \sum_{n=1}^\infty\frac{\left(x-3\right)^{n}}{n} $$

Solution:

Let us suppose that:

$$ C_{n}=\frac{\left(x-3\right)^{n}}{n} $$

The above series will converge for x = 3. Now, for manual computation, we have to use the ratio test.

$$ L= \lim_{n \to \infty}\frac{\left(x-3\right)^{n}}{n} $$

$$ L= \lim_{n \to \infty}[\frac{\left(x-3\right)^{n+1}}{n+1}* \frac{n}{\left(x-3\right)^n}] $$

$$ L= \lim_{n \to \infty}[\frac{\left(x-3\right)^{∞+1}}{∞+1}* \frac{∞}{\left(x-3\right)^∞}] $$

$$ L=\lim_{n \to \infty}[\frac{\left(x-3\right)^{1}}{1}* \frac{∞}{\left(x-3\right)}] $$

$$ \left|x-3\right| $$

Now, this series will only converge if x-3 < 1. Otherwise, for x-3 > 1, the series diverges. So, the radius of convergence is 1. Now, by taking any of the above inequalities, we can determine the interval of convergence.

$$ \left|x-3\right|≤1 $$

$$ -1<\left|x-3\right|<1 $$

$$ -1+3 $$

FAQs:

Can radius of convergence be zero?

When the given series converges at a single point, then we can say that the radius of convergence is zero. Since the convergence happens at a single point, the radius of convergence calculator indicates this by finding the series converges for a single value. This means the series diverges for any non-zero values away from that point.

What is the radius of convergence if the limit is zero?

When the upper limit tends to zero, the radius of convergence extends to infinity. If the limit is a finite positive number, the radius of convergence can be obtained by taking the inverse of the limit superior.

Can we calculate an infinite radius of convergence?

We can only calculate the radius of convergence to be infinite if the series converges for all complex numbers z.