Convergence Tests Summary

Part 1 of 7 — The Master Chart

All Tests at a Glance

Which Test? 🎯

Key Takeaways — Part 1

Always start with the divergence test, then choose the appropriate test.

Convergence Tests

Part 2 of 7 — Decision Flowchart

Step-by-Step Strategy

1. Is $a_n o 0$? If not → diverges ✅
2. Is it geometric ($ar^n$)? → Check $|r|$
3. Is it $p$-series ($1/n^p$)? → Check $p > 1$
4. Is it alternating? → AST
5. Has factorials or $n$th powers? → Ratio/Root test
6. Resembles known series? → Comparison/Limit Comparison
7. Can write $a_n = f(n)$ nicely? → Integral test

Flowchart Practice 🎯

Key Takeaways — Part 2

Follow the flowchart: divergence test first, then identify the series type.

Convergence Tests

Part 3 of 7 — Common Pitfalls

Mistakes to Avoid

1. Divergence test ≠ convergence test: $a_n o 0$ does NOT prove convergence
2. Ratio test $L = 1$: INCONCLUSIVE (try another test!)
3. Comparison wrong direction: bigger than convergent → NOT helpful
4. Forgetting to check AST conditions: must verify decreasing AND $o 0$

Pitfalls 🎯

Key Takeaways — Part 3

The divergence test can only prove divergence, never convergence.

Convergence Tests

Part 4 of 7 — Mixed Practice I

Determine Convergence 🎯

Practice Complete!

Convergence Tests

Part 5 of 7 — Mixed Practice II

More Practice 🎯

More Practice Done!

Convergence Tests

Part 6 of 7 — AP Exam Strategies

Free Response Tips

1. State the test you're using ("By the Ratio Test...")
2. Show the limit computation
3. State the conclusion with the value of $L$
4. For absolute/conditional: test absolute first, then alternating if needed

AP Format Practice 🎯

Key Takeaways — Part 6

Always explain which test and show your work on the AP exam.

Convergence Tests — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Convergence Tests — Complete! ✅