Question

What is the difference between radius of convergence and interval of convergence?

Answer 1

Hint: The terms radius of convergence and interval of convergence are related to the power series. We will first recall the definitions of radius of convergence and interval of convergence. Then we will discuss the differences between the two to get the desired answer.

Complete step by step answer:
We have to write the difference between radius of convergence and interval of convergence.
We know that in mathematics convergence is the property of the function to increase or decrease as the number of terms in the series gets increased.
The radius of convergence of a power series is the radius of the largest disk in which the series converges. A power series converges in a symmetric interval about its expansion point.
The difference between the radius of convergence and interval of convergence is that the radius gives us the number of values where the series converges and interval gives us the value exactly where the series converges.
If a power series converges at $x=a$ the point $x=a$ is interval of convergence and radius of convergence is zero.

Note: To find the interval of convergence first we need to find the radius of convergence. To find the radius of convergence we need to do the ratio test of the series and then calculate the radius of convergence in three cases i.e. if limit is zero, if limit is equal to $N\times \left| x-a \right|$ and if limit is $\infty $.