cal 3 Test 4

What should you ALWAYS check first?

nth-Term Test: Calculate lim aₙ. If the limit isn't 0, the series diverges.

What test should you use for arⁿ⁻¹?

Geometric Series Test: Converges if |r| < 1; diverges if |r| ≥ 1.

What test applies to 1/nᵖ?

p-Series Test: Converges if p > 1; diverges if p ≤ 1.

What test applies when signs alternate?

Alternating Series Test: Terms must decrease and approach 0.

When should you consider the Integral Test?

When terms are positive, continuous, decreasing, and easy to integrate.

When should you use Direct Comparison?

When you can compare positive terms using an inequality to a known series.

When should you use Limit Comparison?

When a complicated positive fraction resembles a known series.

When should you use the Ratio Test?

When you see factorials, exponentials, or complicated consecutive terms.

When should you use the Root Test?

When the entire term is raised to the nth power.

What if a series has negative terms?

Try Ratio or Root Test. If signs alternate, consider Alternating Series Test.