How to Use This Problem Set
Series convergence questions are rarely about performing a long calculation. The real skill is recognizing which convergence test fits the structure of the series.
Before looking at the answer, try to identify the test you would use. On the AP exam, being able to explain why a particular test applies is often just as important as getting the final conclusion.
- Check the terms first. If the terms do not approach zero, use the Divergence Test.
- Look for a geometric series. A constant ratio usually makes this the fastest test.
- Look for alternating signs. Check the Alternating Series Test.
- Compare with a familiar positive series. Use Direct Comparison or Limit Comparison.
- Look for an integral-friendly expression. The Integral Test may be appropriate.
- Look for factorials or powers involving n. The Ratio Test is often efficient.
- Look for an nth power. The Root Test may simplify the problem immediately.
Essential Tests to Know
Divergence Test
If
then the series
must diverge. Remember that the converse is not true: if the terms approach zero, the series may either converge or diverge.
Geometric Series
A geometric series
converges when
and diverges when .
p-Series
The benchmark series
converges when and diverges when .
Comparison Tests
For positive-term series, if and converges, then also converges.
Limit Comparison is especially useful when two expressions have the same dominant behavior:
Alternating Series Test
An alternating series
converges if the terms eventually decrease and .
Ratio Test
If , the series converges absolutely. If , it diverges. If , the test is inconclusive.
Root Test
As with the Ratio Test, a value below 1 gives absolute convergence, a value above 1 gives divergence, and a value equal to 1 is inconclusive.
Worked Warm-Up Problems
Determine whether
converges or diverges.
Determine whether
converges or diverges.
Determine whether
converges or diverges.
Practice Problem Set
For each problem, determine whether the series converges or diverges. When appropriate, identify the specific convergence test you would use.
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges absolutely, conditionally, or diverges.
Show answer
Determine whether converges absolutely, conditionally, or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges. Choose the most efficient test.
Show answer
Determine whether converges or diverges. Choose the most efficient test.
Show answer
Determine whether converges or diverges. Choose the most efficient test.
Show answer
Determine whether converges or diverges. Choose the most efficient test.
Show answer
Challenge Problems
Determine whether converges or diverges.
Show answer
Determine whether converges or diverges.
Show answer
Determine whether converges absolutely, conditionally, or diverges.
Show answer
Common Mistakes
The condition is necessary for convergence, but it is not sufficient. The harmonic series is the classic counterexample: its terms approach zero, but the series diverges.
The Ratio Test is powerful, but it is not always the fastest method. A geometric series, p-series, or obvious comparison may be much easier.
Alternating signs alone are not enough. The terms must approach zero, and the magnitudes must eventually decrease for the Alternating Series Test to apply.
If the limit comparison is a finite positive number, the two positive-term series have the same convergence behavior. The actual value of that limit does not need to be 1.
If the Ratio Test produces , the test is inconclusive. You must switch to another method.
AP Calculus BC Test-Selection Checklist
- Geometric? Find the common ratio.
- Terms fail to approach zero? Use the Divergence Test.
- Looks like ? Think p-series.
- Positive terms resembling a familiar series? Try Comparison or Limit Comparison.
- Alternating signs? Check the Alternating Series Test and then consider absolute convergence.
- Factorials or products? Consider the Ratio Test.
- Everything raised to the nth power? Consider the Root Test.
- Easy continuous function? Consider the Integral Test.
Final Takeaway
The hardest part of series convergence problems is usually not the algebra. It is recognizing the structure of the series.
A strong AP Calculus BC student should be able to look at a new series and quickly ask: "What does this resemble?" That recognition determines the most efficient test.
When you practice, do not just record whether a series converges or diverges. Record which test you chose and why. That is the skill that transfers directly to unfamiliar AP exam questions.
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