September 6, 2011 15:41 c09 Sheet number 50 Page number 645 cyan magenta yellow black Summary of Convergence Tests name Divergence Test (9.4.1) Integral Test (9.4.4) statement If lim uk ≠ 0, then comments If lim uk = 0, then k→ + ∞ may not converge. uk diverges. k→+ ∞ Let uk be a series with positive terms. If f is a function that is decreasing and continuous on an interval [a, + ∞) and such that uk = f (k) for all k ≥ a, then ∞ k =1 uk +∞ and uk may or This test only applies to series that have positive terms. Try this test when f (x) is easy to integrate. f (x) dx a both converge or both diverge. ∞ Let a and k=1 k terms such that Comparison Test (9.5.1) ∞ b k=1 k be series with nonnegative a1 ≤ b1, a2 ≤ b2, . . . , ak ≤ bk , . . . If bk converges, then ak converges, and if diverges, then bk diverges. Let ak and ak bk be series with positive terms and let a r = lim k k→ + ∞ bk Limit Comparison Test (9.5.4) If 0 < r < + ∞, then both series converge or both diverge. Let Ratio Test (9.5.5) Root Test (9.5.6) Try this test as a last resort; other tests are often easier to apply. This is easier to apply than the comparison test, but still requires some skill in choosing the series bk for comparison. uk be a series with positive terms and suppose that r= uk+1 k→+ ∞ uk lim (a) Series converges if r < 1. (b) Series diverges if r > 1 or r = + ∞. (c) The test is inconclusive if r = 1. Let This test only applies to series with nonnegative terms. Try this test when uk involves factorials or kth powers. uk be a series with positive terms and suppose that r= lim k→+ ∞ k uk (a) The series converges if r < 1. (b) The series diverges if r > 1 or r = + ∞. (c) The test is inconclusive if r = 1. Try this test when uk involves kth powers. If ak > 0 for k = 1, 2, 3, . . . , then the series Alternating Series Test (9.6.1) a1 − a 2 + a 3 − a 4 + . . . −a1 + a2 – a3 + a4 − . . . converge if the following conditions hold: (a) a1 ≥ a2 ≥ a3 ≥ . . . (b) lim ak = 0 This test applies only to alternating series. k→+ ∞ Let Ratio Test for Absolute Convergence (9.6.5) uk be a series with nonzero terms and suppose that | uk +1 | k→+ ∞ | uk | r = lim (a) The series converges absolutely if r < 1. (b) The series diverges if r > 1 or r = + ∞. (c) The test is inconclusive if r = 1. The series need not have positive terms and need not be alternating to use this test.