Test Prep. - STRATEGIES FOR F TESTIN G SERIES The Basic...

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F STRATEGIES FOR TESTING SERIES W e have considered many basic convergence tests for inFnite series in Chapter 10. The Basic Tests for InFnite Series (A) Divergence Test (B) Integral Test (C) Comparison Test (D) Limit Comparison Test (E) Ratio and Root Tests (±) Leibniz Test for Alternating Series So given a particular inFnite series, which test should you use? There is no single answer, but there are some facts and general guidelines you should keep in mind. ±irst of all, to develop your intuition, you should be familiar with the following key examples: Geometric series: X n = 0 r n converges if | r | < 1 and diverges otherwise. p -series: X n = 1 1 n p converges if p > 1 and diverges otherwise. X n = 0 1 n ! converges by the Ratio Test. X n = 1 ( 1 ) n n converges conditionally but not absolutely (Example 4, Section 10.4). A Frst step, when testing an inFnite series X n = 1 a n is to check that the general term a n approaches zero. By the Divergence Test, if lim n →∞ a n does not exist, or if lim n →∞ a n exists but is not equal to zero, then X a n diverges. EXAMPLE 1 First Check Whether the General Term Approaches Zero Determine the convergence of (a) X n = 0 n 2 n 2 + 1 and (b) X n = 0 ( 1 ) n n 2 n 2 + 1 . Solution Both series diverge: REMINDER The Leibniz Test states that X n = 0 ( 1 ) n a n converges if the sequence { a n } is positive, decreasing, and lim n →∞ a n = 0 . This test does not apply to the series in Example 1 (b) because the terms a n = n 2 n 2 + 1 do not satisfy either of the two hypotheses: { a n } is not decreasing and lim n →∞ a n is nonzero. (a) lim n →∞ n 2 n 2 + 1 = lim n →∞ 1 1 + n 2 = 1 (nonzero) X n = 0 n 2 n 2 + 1 diverges (b) lim n →∞ ( 1 ) n n 2 n 2 + 1 = lim n →∞ ( 1 ) n 1 + n 2 (does not exist) X n = 0 ( 1 ) n n 2 n 2 + 1 diverges Next, ask yourself if the series resembles a series whose behavior is known. If so, try using the Comparison or Limit Comparison Tests. The following guidelines may be useful: F1
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F2 APPENDIX F STRATEGIES FOR TESTING SERIES If the series contains: Try: n k or other powers of n Comparison or Limit Comparison with p -series b n ( b constant) Ratio Test factorials such as n ! , (2n)! Ratio Test n n Root Test ( 1 ) n Check for absolute convergence or use Leibniz Test EXAMPLE 2 Check for a Simple Comparison Determine the convergence of X n = 1 1 1 . 6 n + n 1 Solution The series converges by the Comparison Test because 1 1 . 6 n + n 1 1 1 . 6 n and X n = 1 1 1 . 6 n is a convergent geometric series µ with r = 1 1 . 6 . Although the inequal-
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Test Prep. - STRATEGIES FOR F TESTIN G SERIES The Basic...

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