Cogito
AP Calculus BC · Chapter 4 · Lesson 1
Choosing a Convergence Test
Which tool for which series.
11 problems · about 26 minutes · AP Calculus BC 10.3, 10.5, 10.8
What this lesson teaches
The student selects and applies an appropriate convergence test.
- The nth term test can only prove divergence.
- A geometric series converges when |r| < 1; a p-series when p > 1.
- The ratio test is decisive except when its limit is exactly 1.
Warm Up
Straightforward practice. Get the method working first.
4 problemsSum of 1/n^0.5. Converge? 1 for yes, 0 for no.
Answer 0
Why Diverges.
Sum of 1/n⁴. Converge? 1 for yes, 0 for no.
Answer 1
Why p = 4.
Geometric with r = 0.9. Converge? 1 for yes, 0 for no.
Answer 1
Why Under 1.
Geometric with r = 1.1. Converge? 1 for yes, 0 for no.
Answer 0
Why Above 1.
Build It Up
The same ideas with more to keep track of.
3 problemsRatio test limit is 0.4. Converge? 1 for yes, 0 for no.
Answer 1
Why Below 1.
Ratio test limit is exactly 1. Is the test decisive? 1 for yes, 0 for no.
Answer 0
Why It is silent there.
Sort each series by whether it converges.
Answer Converges: Sum of 1/n², Geometric, r = 1/2 · Diverges: Sum of 1/n, Sum of n
Why Check p, then the ratio, then the terms.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsTerms approach 3, not 0. Converge? 1 for yes, 0 for no.
Answer 0
Why The nth term test.
Match each series to the test that settles it most directly.
Answer Terms approach 5 → nth term test; Sum of 1/n^1.5 → p-series; Terms with factorials → Ratio test
Why Factorials cancel beautifully in a ratio.
A p-Series: Sum of 1/n³. Does it converge? 1 for yes, 0 for no.
Answer 1
Why Converges.
The Borderline: Sum of 1/n. Does it converge? 1 for yes, 0 for no.
Answer 0
Why Diverges.