Radius of Convergence Calculator
Free online Radius of Convergence Calculator tool evaluates the radius of a convergence of a power series. Simply enter your function and variable range in the given input sections and tap on the calculate button to get the instant output along with a detailed procedure.
Radius of Convergence Calculator: If you want to know the radius of convergence of a power series equation and need any help? Then we are here you to assist for any kind of math solutions. Have a look at the Radius of Convergence Calculator to solve the power series function within seconds. This article gives a detailed description of steps to solve the radius of convergence manually and we will explain it with few examples.
Method to Calculate Radius of Convergence of a Power Series
Follow these simple steps to find out the radius of convergence of a power series
- Take a power series
- Consider the value of x for which the power series will converge
- To get the radius of convergence, find out ratio test
- And evaluate the function as per the ratio test
- Ratio test will gives you the limit value
- Substitute the limit value to get the R i.e Radius of Convergence
Example
Question: Find the Radius of Convergence for the power series Sigma n=to infinity 2n/nx(4x-8)n
Solution:
Let us take Cn=2n/nx(4x-8)n
We know that this power series will converge for x=2
For the above power series, the ratio test will be
L=Cn+1/Cn
L=lim n to infinity 2n+1(4x-8)n+1/n+1*n/2n(4x-8)n
lim n to infinity 2n(4x-8)/n+1
(4x-8) lim n to infinity 2n/n+1
=2(4x-8)
So we will get the below convergence info from this
2(4x-8)<1
x-a
8(x-2)<1
(x-2)<⅛
8(x-2)>1
(x-2)>1/8
So, the radius of convergence for the power series is R=1/8
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FAQs on Radius of Convegence
1. What is the Radius of Convergence?
Radius of Convergence of a power series is the radius of the largest disk in which the series converges. It will be non negative real number or infinity. In the positive case, the power series converges absolutely.
2. What is the radius of convergence is 0?
The radius of convergence R =0 tells that the distance between the center of a power series interval of convergence and its endpoints.
3. Can the radius of convergence be negative?
No, the radius of convergence can never be a negative number.
4. What is the ratio test for convergence?
The ratio test defines that: if L<1 then the series is convergent or if L>1 then the series is divergent. In case L=1, tes is inclusive, because it satisfies both convergent and divergent.