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Cogito

AP Calculus BC · Chapter 3 · Lesson 3

Practice Worksheet

The nth Term Test and p-Series

The first check, and a family worth memorising.

12 problems · about 21 minutes · LIM-7.A

NameDate

Figure — use these to answer the problems

The terms of a p-series look like this curve. They shrink toward zero, which is necessary but not sufficient.
-8-8-6-6-4-4-2-222446688
y = 1/x + 0
  • Point(3, 0.33)
a1
Point on the curve3
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. p = 2.5. Does the p-series converge? 1 yes, 0 no.

    Answer

  2. What can the nth term test establish?

    • Divergence only, when the terms do not approach zero.
    • Both convergence and divergence.

  3. p = 3. Does the p-series converge? 1 yes, 0 no.

    Answer

  4. p = 1. Does it converge? 1 yes, 0 no.

    Answer

  5. p = 0.5. Does it converge? 1 yes, 0 no.

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. The terms approach 0. Does the series converge? 1 yes, 0 no.

    Answer

  2. The nth term is 2n ÷ (n + 5). Limit as n grows?

    Answer

  3. That series therefore does what? 1 converges, 2 diverges.

    Answer

Stretch Yourself

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

4 problems

  1. Sort each p value by what the p-series does.

    Write each item under the heading it belongs to: p = 2 · p = 1.5 · p = 1 · p = 0.5

    Converges

    Diverges

  2. p = 4. Does the p-series converge? 1 yes, 0 no.

    Answer

  3. The Limit

    The nth term is n ÷ (n + 1). What is its limit as n grows without bound?

    Answer

  4. The p-Series

    The series summing 1 ÷ n². Does it converge? 1 yes, 0 no.

    Answer