Classifying Infinite Series
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The document discusses convergence testing on the AP Calculus BC exam. It states that series problems will appear on both the multiple choice and free response sections. Students must be able to classify series as convergent or divergent using the tests learned in class. It then provides examples of series classification problems and the justification for each classification using a specific convergence test.
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- 1. CONVERGENCE TESTING AP CALCULUS BC EXAM REVIEW
- 2. SERIES ON THE AP EXAM • SERIES WILL BE TESTED ON BOTH THE MULTIPLE CHOICE AND THE FREE RESPONSE SECTIONS OF THE EXAM. • BEING ABLE TO CLASSIFY A SERIES AS BEING CONVERGENT OR DIVERGENT USING THE TESTS WE LEARNED IN THE COURSE WILL BE ESSENTIAL. • FOR EACH OF THE SERIES ON THE FOLLOWING SLIDES, CLASSIFY IT AS BEING CONVERGENT OR DIVERGENT AND MAKE SURE YOU CAN JUSTIFY YOUR CHOICE. YOU SHOULD BE ABLE TO CITE A SPECIFIC TEST FOR EACH ONE.
- 3. CONVERGENT OR DIVERGENT? WHY? 𝑛=1 ∞ 𝑛2 + 1 𝑛
- 4. DIVERGENT! • LIM 𝑛→∞ 𝑛2+1 𝑛 = ∞ • SINCE THE LIMIT DOES NOT EQUAL 0, THIS SERIES DIVERGES BY THE NTH-TERM TEST.
- 5. CONVERGENT OR DIVERGENT? WHY? 𝑛=1 ∞ 1 𝑛 𝜋
- 6. CONVERGENT! • THIS IS A P-SERIES. • SINCE P (WHICH IS PI) IS GREATER THAN 1, THE SERIES WILL CONVERGE.
- 7. CONVERGENT OR DIVERGENT? WHY? 𝑛=1 ∞ 2 𝑛+1 5 𝑛
- 8. CONVERGENT! • THIS IS A GEOMETRIC SERIES, WHICH WE CAN SEE MORE CLEARLY IF WE REWRITE IT AS 𝑛=1 ∞ 2 2 5 𝑛 . • SINCE THE COMMON RATIO OF 2/5 IS LESS THAN 1, THIS IS A CONVERGENT GEOMETRIC SERIES.
- 9. CONVERGENT OR DIVERGENT? WHY? 𝑛=1 ∞ −1 𝑛 𝑛 𝑛 + 1
- 10. CONVERGENT! • THIS IS AN ALTERNATING SERIES, AND SINCE THE SEQUENCE PART OF THE SERIES IS ALWAYS POSITIVE, IS ALWAYS DECREASING, AND IS HEADED TOWARDS 0, THE SERIES CONVERGES. • WE COULD FURTHER CLASSIFY THIS ONE AS BEING CONDITIONALLY CONVERGENT BECAUSE THE SEQUENCE AS ITS OWN SERIES WOULD DIVERGE (BY COMPARISON TO A P-SERIES WHERE P = ½).