Our radius of convergence calculator uses the ratio test or the root test to calculate the radius of convergence and interval of convergence for which the power series converges.
“The distance from the center point of the series to the nearest point where the series converges”.
$$ R = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| $$
Where:
The range of series diverges the limit of the absolute value in the power series by taking into account the radius of the convergence calculator.
The radius of convergence can be calculated using a variety of methods, but here we discuss basic two:
The method involves taking the limit of value in the power series to find the convergence, divergence, interval, and radius of convergence.
$$ \lim_{n\to\infty} \left|\frac{x^{n+1}}{x^n}\right| $$
Numeric Calculation:
Assume the power series $$ \sum_{n=0}^∞ x^n $$ at which the center of the series is a = 0, to calculate the radius of convergence, we can use the ratio test. Taking the ratio of successive terms, we get:
$$ \lim_{n\to\infty} \left| \frac{x^{n+1}}{x^n} \right|=|x| $$
$$ R = \limsup_{n\to\infty} \sqrt[n]{|a_n|} $$
This is another common method for calculating the radius of convergence. It involves taking the limit of the $n$th root of the absolute value of the $n$th term in the Taylor series.
Numeric Calculation:
Assume the power series $$ \sum_{n=0}^∞ x^n $$ at which the center of the series is a = 0, to calculate the radius of convergence, we can use the root test as well as radius of convergence calculator. We get:
$$ \sqrt[n]{|x^n|} = |x| $$
Example:
Find the radius of convergence when $$ \left|x - 5\right| \le 1 $$
$$ \sum_{n=1}^\infty\frac{\left(x-5\right)^{n}}{n} $$
The radius and interval of convergence calculator utilizes the ratio test method for the determination of the radius of convergence as we know this test has a formula as follows:
$$ L=\lim_{n \to \infty}\left|\dfrac{a_{n+1}}{a_n}\right| $$
$$ \left|L\right| \le 1 $$
$$ a_n=\frac{\left(x - 5\right)^{n}}{n} $$
$$ L=\lim_{n \to \infty}\left|\dfrac{\frac{\left(x - 5\right)^{n+1}}{n+1}}{\frac{\left(x - 5\right)^{n}}{n}}\right| $$
$$ L=\lim_{n \to \infty}\left|\frac{n \left(x - 5\right)^{- n} \left(x - 5\right)^{n + 1}}{n + 1}\right| $$
$$ L=\left|x - 5\right| $$
$$ \left|x - 5\right| \le 1 $$
The ratio test calculator integrates mathematical formulas to calculate the radius of convergence of a power series. It transforms the complex analysis of given equations into an accessible process by making convergence assessments.
Our power series convergence calculator will determine the following:
Function/Power Series |
Interval of Convergence |
$$ \sin(x)=\sum^\infty_{n=0}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!} $$ |
$$ \mathbb{R} $$ |
$$ \cos(x)=\sum^\infty_{n=0}\frac{(-1)^{n}x^{2n}}{(2n)!} $$ |
$$ \mathbb{R} $$ |
$$ \tan^{-1}(x)=\sum^\infty_{n=0}\frac{(-1)^{n}x^{2n+1}}{2n+1} $$ |
$$ |x|<1 $$ |
$$ e^x=\sum^\infty_{n=0}\frac{x^n}{n!} $$ |
$$ \mathbb{R} $$ |
$$ \frac{1}{1-x}=\sum^\infty_{n=0}x^n $$ |
$$ |x|<1 $$ |
$$ \frac{1}{1+x}=\sum^\infty_{n=0}(-1)^{n}x^n $$ |
$$ |x|<1 $$ |
$$ \ln(x)=\sum^\infty_{n=1}\frac{(-1)^{n}(x-1)^n}{n} $$ |
$$ |1-x|<1 $$ |
$$ \ln(1+x)=\sum^\infty_{n=1}\frac{(-1)^{n+1}x^n}{n} $$ |
$$ |x|<1 $$ |
$$ \ln(1-x)=-\sum^\infty_{n=1}\frac{x^n}{n} $$ |
$$ |x|<1 $$ |
$$ \ln(\frac{1+x}{1-x})=2\sum^\infty_{n=0}\frac{x^{2n+1}}{2n+1} $$ |
$$ |x^2|<1 $$ |
Wikipedia: Radius of convergence, Radius of convergence in complex analysis, Convergence on the boundary, Abscissa of convergence of a Dirichlet series.
Lumen Learning: Form and Convergence of a Power Series, Root Test, The Integral and Comparison Tests, Alternating Series, Taylor and Maclaurin Series.
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