Tests for Convergence and Divergence

1. Tests for Convergence and Divergence

2. If Then the series diverges This does not work in reverse! Just because does not mean that the series converges. When To Use Always check first nth term test Warning! Examples: Diverges because

3. Geometric series test Only use for geometric series… duh converges Make sure that before using converges to In If Then the series converges, otherwise the series diverges.

4. Telescoping Series • This is used to find the sum of the series, not necessarily if it converges or diverges. • You might need to use partial fractions. • Write out the first few terms to see what “telescopes” out. • Find the limit of Sn as n  ∞

5. The value of the Integral is not Necessarily the Value to which The series converges Warning! Rarely used except to prove P-test Integral Test Convert to the improper integral Example:

6. P-test Usually only used in comparison tests Only use on series in the form: Converges if p > 1 Diverges if 0 < p < 1 Examples:

7. Comparison Test series converges series diverges Use primarily for if you have an an that is less than a divergent or more than a convergent the comparison test is inconclusive Examples: When to Use: types of series Therefore series converges

8. use when you have an effective power in the denominator Limit Comparison Test Therefore the series converges because converges by P-test Examples:

9. Absolute Convergence Theorem

10. Ratio Test Good for mix of factorials, exponentials, powers The series does not necessarily converge to ρ!!! If ρ = 1, you must use a different test If ρ < 1, series converges If ρ > 1, series diverges If ρ = 1, inconclusive Therefore series converges

11. Root Test Good when you have something complicated to the nth power The series does not necessarily converge to ρ If ρ < 1, series converges If ρ > 1, series diverges If ρ = 1, inconclusive Therefore series converges

12. Alternating Series Test

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