ratio or root test
ΣCALCULUS II STUDY GUIDE
Series and Convergence Tests
Select comparison, ratio, root, integral, alternating, and telescoping tests for infinite series.
8 key ideas8 flashcards5 practice questions
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Match the term pattern to a test that can decide it.
comparison or integral test
absolute test, then alternating test
Key ideas to know
Start with the relationships between ideas. Then close the notes and explain each one from memory.
- 01
An infinite series converges when its partial sums approach a finite number.
- 02
If aₙ does not tend to zero, the series Σaₙ diverges.
- 03
A geometric series converges when the absolute ratio is below one.
- 04
A p-series Σ1/nᵖ converges exactly when p is greater than one.
- 05
Comparison tests work best for nonnegative terms resembling known benchmark series.
- 06
The ratio test is useful when factorials or repeated powers appear.
- 07
The alternating series test needs decreasing magnitudes that tend to zero.
- 08
Absolute convergence implies convergence, while conditional convergence does not imply absolute convergence.
PART OF THE CALCULUS II GUIDE
See every set in this course and follow a focused review order.
Open the full Calculus II guide →Two ideas worth correcting now
NOT QUITE
→Terms tending to zero prove series convergence
USE THIS INSTEAD
This condition is necessary but not sufficient.
NOT QUITE
→The ratio test settles every series
USE THIS INSTEAD
A ratio limit of one gives no decision.
Flashcards
Answer before opening each card. The effort to retrieve is part of the learning.
1What sequence determines series convergence?Show answer ＋
The sequence of partial sums.
2What does the nth-term test prove when aₙ does not tend to zero?Show answer ＋
The series diverges.
3When does Σarⁿ converge?Show answer ＋
When |r| < 1.
4When does Σ1/nᵖ converge?Show answer ＋
When p > 1.
5What result makes ratio testing decisive?Show answer ＋
A limiting absolute ratio below one gives absolute convergence; above one gives divergence.
6When is the ratio test inconclusive?Show answer ＋
When its limit equals one.
7What does alternating-series error not exceed?Show answer ＋
The magnitude of the first omitted term.
8What is conditional convergence?Show answer ＋
The signed series converges while the absolute series diverges.
Explain it in your own words
Use the answer as a check after you have written or spoken your response.
01Why does aₙ → 0 not prove Σaₙ converges?
The harmonic series has terms tending to zero but divergent partial sums.
02Which test fits Σn!/3ⁿ?
The ratio test, because factorial and exponential factors simplify after dividing successive terms.
03Which benchmark fits terms behaving like 1/n²?
The convergent p-series with p = 2.
04How does a telescoping series work?
Most terms cancel in its partial sum, leaving only boundary terms.
05How is an alternating sum estimated to a requested accuracy?
Choose enough terms so the first omitted magnitude is below the error tolerance.
Related study materials
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