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Cogito

AP Calculus BC · Chapter 5 · Lesson 2

Practice Worksheet

Radius and Interval of Convergence

Where a power series is allowed to live.

12 problems · about 22 minutes · LIM-8.D

NameDate

Figure — use these to answer the problems

The interval of convergence is a band around the centre. Move the marker and picture the radius either side.
-10-8-6-4-20246810

0

Centre of the series0
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Centred at 0 with radius 6. What is the right endpoint?

    Answer

  2. Why must endpoints be tested separately?

    • The ratio test gives exactly 1 there, so it is inconclusive.
    • They are always included.

  3. Converges for |x| < 3. Radius?

    Answer

  4. Centred at 0 with radius 4. What is the right endpoint?

    Answer

  5. Centred at 0 with radius 4. Left endpoint?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Centred at 3 with radius 2. Right endpoint?

    Answer

  2. Is the ratio test conclusive at an endpoint? 1 yes, 0 no.

    Answer

  3. A radius of 0. At how many points does the series converge?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Put the process for finding an interval in order.

    Write 1 to 4 in the boxes to put these in order.

    • Apply the ratio test to find the radius.
    • Write the open interval around the centre.
    • Substitute each endpoint separately.
    • Test each resulting numerical series and include or exclude it.

  2. Centred at 1 with radius 3. Left endpoint?

    Answer

  3. The Radius

    A series centred at 0 converges for |x| < 5. What is the radius of convergence?

    Answer

  4. The Endpoints

    How many endpoints must be tested separately?

    Answer