Standard Tests for evaluating convergence or divergence of Infinite Series of Constant Terms

B = applies to both positive & alternating series P = only applies to positive term series

A = only applies to alternating series

B	nth term test - If lim an ≠ 0 or does not exist, series diverges . n → ∞ for divergence
P	Integral test - If ∑ an is a positive series and f(x) is positive valued decreasing, continuous function for x ≥1. If f(n) = an for all integers n ≥ 1, then the series and the improper integral ∞ ∑ an and ∫ f(x)dx 1 either both diverge or both converge The integral must have a finite value for convergence.
P	Comparison Test - Suppose ∑ an and ∑ bn are positive term series. Then a. ∑an converges if ∑ bn converges and an ≤ bn for all n. b. ∑an diverges if ∑ bn diverges and an ≥ bn for all n.
P	Limit Comparison Test - Suppose that ∑ an and ∑bn are positive term series. If lim ( an/ bn ) exists and 0 < L < +∞ , n → ∞ then either both series diverge or both series converge. This test fails if the limit equals 0 or +∞ .
A	If the alternating series, ∑(-1)^n+1 an satisfies the two conditions: a. an ≥ an+1 > 0 for all n and b. lim an = 0 n → ∞ then the alternating series converges. Else, this test does not apply. An alternating series that converges but not absolutely is conditionally convergent. For example: ∑ (-1)^n an converges but ∑ an diverges
B	Ratio Test - If P= lim | (an+1)/ an | exists, then the infinite the series n → ∞ ∑ an converges absolutely if P < 1, diverges if P > 1 (or P = ∞) If P = 1 this test fails.
B	Root Test – Suppose P = lim |an| ^1/n exists or is infinite, then n → ∞ the series ∑ an a. Converges absolutely if P < 1 b. Diverges if P > 1 c. Test fails if P = 1

Return to Calculus 2 Notes Involving Sequences, Infinite Series, etc

Copyright © 2017 Richard C. Coddington

All rights reserved.