Cogito
AP Calculus BC · Chapter 3 · Lesson 3
The nth Term Test and p-Series
The first check, and a family worth memorising.
12 problems · about 21 minutes · LIM-7.A
What this lesson teaches
The student applies the nth term test for divergence and classifies p-series.
- If the terms do not approach zero, the series diverges.
- Terms approaching zero proves nothing on its own.
- A p-series converges exactly when p > 1.
Warm Up
Straightforward practice. Get the method working first.
5 problemsp = 2.5. Does the p-series converge? 1 yes, 0 no.
Answer 1
Why Yes.
What can the nth term test establish?
Answer Divergence only, when the terms do not approach zero.
Why Divergence only.
p = 3. Does the p-series converge? 1 yes, 0 no.
Answer 1
Why Above 1.
p = 1. Does it converge? 1 yes, 0 no.
Answer 0
Why The harmonic series.
p = 0.5. Does it converge? 1 yes, 0 no.
Answer 0
Why Below 1.
Build It Up
The same ideas with more to keep track of.
3 problemsThe terms approach 0. Does the series converge? 1 yes, 0 no.
Answer 0
Why Enter 0 for "not necessarily".
The nth term is 2n ÷ (n + 5). Limit as n grows?
Answer 2
Why 2 ÷ 1.
That series therefore does what? 1 converges, 2 diverges.
Answer 2
Why The terms do not reach zero.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each p value by what the p-series does.
Answer Converges: p = 2, p = 1.5 · Diverges: p = 1, p = 0.5
Why The boundary belongs to the divergent side.
p = 4. Does the p-series converge? 1 yes, 0 no.
Answer 1
Why Above 1.
The Limit: The nth term is n ÷ (n + 1). What is its limit as n grows without bound?
Answer 1
Why 1.
The p-Series: The series summing 1 ÷ n². Does it converge? 1 yes, 0 no.
Answer 1
Why Yes.