Lesson 11 6 Absolute Convergence Ratio Test Root

Lesson 11 -6 Absolute Convergence Ratio Test Root Test

Vocabulary • Absolute Convergent – a series of numbers that alternate in sign, like the summation of the following • Conditionally Convergent -- a convergent series that is not absolutely convergent • Rearrangement – a reordering of terms in an infinite series

Types of Series • Geometric • Telescoping • Harmonic • P-Series • Alternating

Types of Series Tests • Divergence (easiest to check) – if lim an ≠ 0 then it diverges n→∞ – if lim an = 0 then it is inconclusive • Comparison (last resort) – if an ≤ bn for all n 0 then it diverges – if lim an = 0 then it is inconclusive • Limit Comparison (better than comparison) – if lim an ≠ 0 then it diverges – if lim an = 0 then it is inconclusive • Ratio (apply when factorials are involved) • Root (apply when nth power involved)

Ratio Test Let ∑ an be a series an+1 • If lim |----| = L < 1 , then the series ∑a n is n→∞ an absolutely convergent (and therefore convergent) an+1 = L > 1, or lim | ------- | = ∞ , then the |------| • If lim n→∞ an an series ∑an is divergent an+1 ------- | = 1 , the Ratio test is inconclusive (no | • If lim n→∞ an conclusion about the convergence or divergence of the series can be drawn)

Root Test Let ∑ an be a series n • If lim √|an| = L < 1 n→∞ , then the series ∑a n is absolutely convergent (and therefore convergent) n n • If lim √|an| = L > 1 , or lim √|an| = ∞ , then the n→∞ series ∑an is divergent n • If lim √|an| = 1 , the Ratio test is inconclusive (no n→∞ conclusion about the convergence or divergence of the series can be drawn)

11 -6 Example 1 Is the following series convergent or divergent? (-1)n+1 2 n series ------n! Lim bn = indeterminate and L’Hospital’s Rule does not apply! n→∞ n+1 / (n+1)! a 2 n+1 Use Ratio Test, the absolute values lim |----| = lim -------2 n / n! eliminate the (-1) term and we get n→∞ an n! 2 n+1 2 This leads to lim ------n = ------ = 0 therefore it converges (n+1)! 2 n+1

11 -6 Example 2 Is the following series convergent or divergent? series Lim n→∞ 6 n -------3 n - 1 6 n ----3 n – 1 n n = indeterminate and L’Hospital’s Rule does not apply! Ratio test is impossible, so we can try the Root test Lim n→∞ 6 n -----3 n - 1 n 1/n = Lim 6 n ------ = 2 3 n - 1 since p > 1 the series diverges by the Root Test

Homework Pg 745 - 746: problems 3, 4, 7, 11