University:
Amherst College
Course:
MATH 211 | Multivariable Calculus
Academic year:

2024
Views:

320

Pages:

1
Author:

zmysta007u6u

p-Series Test ∞ X 1 1 1 1 The p-series of the form = 1 + p + p + p + ... np 2 3 4 n=1   Converges if p > 1  Diverges if p ≤ 1 USED: For p-series exactly of the form above. Most commonly partnered together with a Comparison Test. NOTE: Using the p-Series Test is a very quick and straightforward justification. Check size of p. WARNING: Be careful to understand the difference between the Geometric Series Test and this p-Series Test. Make sense of the value or purpose of |r| and p for each convergence test. How can that help you memorize each test and the tests’ size arguments? APPROACH: • Recognize the given series in this p-Series form. Notice when the base is changing and the power is a fixed real number. • Pick off the power p. State clearly what the value p equals. • Determine and then state if p is greater than 1 or . . . less or equal to 1. EXAMPLES: Determine and state whether each of the following series Converges or Diverges. Name any Convergence test(s) that you use, and justify all of your work. 1. ∞ X 1 1 1 1 = 1 + 7 + 7 + 7 + ... 7 n 2 3 4 Convergent p-Series with p = 7 > 1. n=1 2. ∞ X 1 1 1 1 = 1 + + + + ... n 2 3 4 Divergent (Harmonic) p-Series with p = 1. n=1 3. ∞ X 1 n=1 4. ∞ X n=1 5. ∞ X n=1 n 5 2 1 n.99 Convergent p-Series with p = Divergent p-Series with p = .99 < 1. ∞ X 1 6 √ =6 1 n n=1 n 2 ∞ ∞ X X 1 5 6. =5 n8 n8 n=1 5 > 1. 2 n=1 Constant Multiple of a Divergent p-Series with p = 1 < 1 is Divergent . 2 Constant Multiple of a Convergent p-Series with p = 8 > 1 is Convergent .

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