Convergence Tests

Convergence tests are essential for deciding whether an infinite series converges. Different series types call for different tools.

主要内容

Tests for positive-term series

1. Comparison test
2. Ratio test (d’Alembert)
3. Root test (Cauchy)
4. Integral test

Tests for alternating series

5. Leibniz test

General-term series

6. Absolute vs. conditional convergence

判别法的选择策略

Steps:

1. Check the necessary condition: verify $\lim_{n \to \infty} a_n = 0$; otherwise it diverges.
2. Identify the type:
   • Positive series: use positive-series tests.
   • Alternating: try Leibniz.
   • General: first test absolute convergence.
3. Pick a tool:
   • Ratio: factorials and exponentials.
   • Root: powers.
   • Comparison: to a known series.
   • Integral: when $a_n = f(n)$ with nice $f$.
4. Special cases:
   • Geometric: use the formula directly.
   • $p$-series: remember $p>1$ converges.
   • Harmonic: diverges.

学习建议

• Master positive-term tests first.
• Know when each test applies (and when it fails, e.g., $\rho = 1$).
• Practice choosing the right test quickly.

Chapters

• Comparison Test
• Ratio Test (d'Alembert)
• Root Test (Cauchy)
• Integral Test
• Leibniz Test
• Absolute vs. Conditional Convergence

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