Calculus II

Calculus II delves into advanced integration techniques, series, and applications of calculus to real-world problems.

Techniques of IntegrationSequences and SeriesParametric and Polar Equations
Power Series and Taylor SeriesConvergence TestsApplications of Integration
Engineering and Physics ApplicationsEconomics and Population ModelsMedicine and Biology
Recognizing the Right TechniqueChecking Your Answers
Techniques of Integr...Sequences and SeriesParametric and Polar...Power Series and Tay...Convergence TestsApplications of Inte...Engineering and Phys...Economics and Popula...Medicine and BiologyRecognizing the Righ...Checking Your Answer...
Advanced Topics

Convergence Tests

Deciding If a Series Adds Up

With infinite series, not all sums make sense—the series might grow forever! Convergence tests help you decide if a series settles down to a specific value or not.

Popular Tests

  • Ratio Test: Looks at the limit of \( |a_{n+1}/a_n| \).
  • Root Test: Uses \( \sqrt[n]{|a_n|} \).
  • Alternating Series Test: For series whose terms alternate in sign.

How It Helps

These tests are critical for knowing when you can use series to approximate real-world quantities or solve equations.

Examples

  • Apply the Ratio Test to \( \sum \frac{n!}{n^n} \) to determine convergence.

  • Use the Alternating Series Test for \( \sum (-1)^n \frac{1}{n} \).

In a Nutshell

Convergence tests let you figure out whether an infinite series has a finite sum.

Next
Applications of Integration