Academy
Radius of Convergence
Level 1 - Math I (Physics) topic page in Power Series.
Principle
The radius of convergence tells how far from the centre a power series can be trusted before it stops converging. For a series centred at \(c\), the main convergence region has the form \(|x-c|\lt R\).
The radius is a distance in the input variable. It does not decide the endpoints \(x=c-R\) and \(x=c+R\); those must be tested separately.
Notation
The value \(R\) may be finite, zero, or infinite. In Level 1 examples it is usually found using the ratio test.
Method
Step 1: Write the nth term
For
the nth term is \(t_n=a_n(x-c)^n\).
Step 2: Apply the ratio test
Use the absolute ratio of consecutive terms:
Step 3: Solve for the distance from the centre
The inequality from the ratio test usually becomes \(|x-c|\lt R\). The number on the right is the radius of convergence.
Rules
If the ratio limit is \(L|x-c|\), then convergence requires \(L|x-c|\lt1\), so \(R=1/L\) when \(L>0\).
Examples
Checks
- The radius is a non-negative distance, not an interval.
- A ratio test result of \(|x-c|\lt R\) says nothing final about endpoints.
- If \(R=\infty\), the series converges for every real \(x\).
- If \(R=0\), the series converges only possibly at its centre.