Cogito
AP Calculus BC · Chapter 5 · Lesson 2
Radius and Interval of Convergence
Where a power series is allowed to live.
12 problems · about 22 minutes · LIM-8.D
What this lesson teaches
The student finds the radius and interval of convergence of a power series.
- The ratio test gives the radius of convergence.
- Each endpoint must be substituted and tested separately.
- A radius may be zero, finite, or infinite.
Warm Up
Straightforward practice. Get the method working first.
5 problemsCentred at 0 with radius 6. What is the right endpoint?
Answer 6
Why 6.
Why must endpoints be tested separately?
Answer The ratio test gives exactly 1 there, so it is inconclusive.
Why The ratio test is inconclusive at the boundary.
Converges for |x| < 3. Radius?
Answer 3
Why The bound.
Centred at 0 with radius 4. What is the right endpoint?
Answer 4
Why Centre plus radius.
Centred at 0 with radius 4. Left endpoint?
Answer -4
Why Centre minus radius.
Build It Up
The same ideas with more to keep track of.
3 problemsCentred at 3 with radius 2. Right endpoint?
Answer 5
Why 3 + 2.
Is the ratio test conclusive at an endpoint? 1 yes, 0 no.
Answer 0
Why The limit is exactly 1 there.
A radius of 0. At how many points does the series converge?
Answer 1
Why Only the centre.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the process for finding an interval in order.
Answer 1. Apply the ratio test to find the radius. 2. Write the open interval around the centre. 3. Substitute each endpoint separately. 4. Test each resulting numerical series and include or exclude it.
Why The endpoints come after the radius.
Centred at 1 with radius 3. Left endpoint?
Answer -2
Why 1 − 3.
The Radius: A series centred at 0 converges for |x| < 5. What is the radius of convergence?
Answer 5
Why 5.
The Endpoints: How many endpoints must be tested separately?
Answer 2
Why 2.