Ratio and Factorial Tests in Series

Mr. Bean introduces the ratio test for convergence in calculus, explaining factorials and exponential properties. He demonstrates the ratio test with examples, highlighting when it is conclusive or inconclusive. The lesson concludes with advanced examples and a summary of key points.

What is the main focus of today's lesson?

• The ratio test for convergence

• The harmonic series

• The integral test

• The p-series test

How is 5 factorial (5!) defined?

• 5 + 4 + 3 + 2 + 1

• 5 - 4 - 3 - 2 - 1

• 5^4

• 5 * 4 * 3 * 2 * 1

What is the result of simplifying n+1 factorial?

• n + 1

• n * (n + 1)

• n + 1 * n!

• n! + 1

How can you rewrite 3^(n+1)?

• 3^n * 3

• 3^(n-1) * 3

• 3^n + 3

• 3^(n+1)

What does the ratio test determine about a series?

• Whether it converges or diverges

• Its exact sum

• Its harmonic nature

• Its rate of divergence

What happens if the ratio test result is less than 1?

• The series is harmonic

• The test is inconclusive

• The series converges

• The series diverges

What is the outcome if the ratio test equals 1?

• The series is geometric

• The series converges

• The series diverges

• The test is inconclusive

Why is the ratio test inconclusive for p-series?

• Because p-series are always divergent

• Because the ratio test always equals 1 for p-series

• Because p-series are always convergent

• Because the ratio test is not applicable to p-series

What should you do if the ratio test is inconclusive?

• Recalculate the ratio

• Assume the series converges

• Assume the series diverges

• Use another test

What is the first step in applying the ratio test?

• Check if the series is alternating

• Calculate the sum of the series

• Find the limit of the ratio as n approaches infinity

• Determine if the series is geometric