A power series converges only for x values near its centre. That set of values is its interval of convergence.
Step 1: Let's Learn
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The radius
The ratio test gives the radius R. The series converges for |x − c| < R and diverges beyond it.
Endpoints need separate work
The ratio test is inconclusive at |x − c| = R. Each endpoint must be substituted and tested on its own.
Four possible intervals
Both endpoints may be included, one, or neither. Only testing them individually decides.
The two extreme cases
A radius of zero means the series converges only at its centre. An infinite radius means it converges everywhere.
The familiar ones
The series for eˣ, sine and cosine all have infinite radius. That is what makes them usable anywhere.
Where a power series is allowed to live
A power series converges on an interval centred at a, of some radius R. Inside it the series represents a function; outside it diverges and represents nothing.
Find R with the ratio test
Apply the ratio test and solve the inequality giving a limit less than 1. The result is |x − a| < R, which reads off the radius directly. This is the standard method for every power series.
Endpoints must be checked separately
The ratio test is silent at the endpoints, so each must be substituted and tested on its own. The interval can be open, closed or half-open, and omitting the endpoint check is a graded loss.
The three possibilities
R can be zero, so the series converges only at the centre; a finite positive number; or infinite, so it converges everywhere. The series for eˣ has infinite radius, which is why it is so useful.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Rosa Find an Interval
Rosa finds the ratio test gives a radius of 2 for a series centred at 0.
- Step 1
A radius of 2 means convergence for |x| < 2.
Step 4: Your Turn
Practice makes it stick.
The Radius
Problem 1 of 2
A series centred at 0 converges for |x| < 5. What is the radius of convergence?
The Endpoints
Problem 2 of 2
How many endpoints must be tested separately?
Find the Interval
1 of 8
Converges for |x| < 3. Radius?
2 of 8
Centred at 0 with radius 4. What is the right endpoint?
3 of 8
Centred at 0 with radius 4. Left endpoint?
4 of 8
Centred at 3 with radius 2. Right endpoint?
5 of 8
Is the ratio test conclusive at an endpoint? 1 yes, 0 no.
6 of 8
A radius of 0. At how many points does the series converge?
7 of 8
Put the process for finding an interval in order.
- 1Write the open interval around the centre.
- 2Substitute each endpoint separately.
- 3Test each resulting numerical series and include or exclude it.
- 4Apply the ratio test to find the radius.
8 of 8
Centred at 1 with radius 3. Left endpoint?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Centred at 0 with radius 6. What is the right endpoint?
Question 2 of 2
Why must endpoints be tested separately?
What You Learned
- The ratio test gives the radius of convergence.
- Each endpoint must be substituted and tested separately.
- A radius may be zero, finite, or infinite.