ENGINEERING MATH II
ENGINEERING MATH II MATH 152
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This 1 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 152 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/226040/math-152-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Date Created: 10/21/15
Tests for Convergence or Divergence of in nite Series l I a w Test for Divergence If Inn ak 0 then 2 ak diverges USE AS A FIRST STEP gt k1 39 00 Geometric Series 2 ark a ar ar2 converges if r lt 1 and sums to 1 37 diverges if Ir 2 1 Telescoping Series Requires difference of terms from partial fractions log rules or problem statement wn39te out rst 3 5 terms 1 pserles 2 P7 converges If pgt1 diverges if plt1 Useful for constructing comparison series w 00 Integral Test Let akfx then both I fxdx and Z ak converge or both diverge f x must satisfy the three conditions 1 k1 39 i positive ii continuous and iii decreasing on the given interval USEFUL IF AK IS EASY T O INTEGRATE OR IF LN APPEARS IN THE SERIES Comparison Test If Zbk converges and bk 3 ak then Zak converges lf Zbk diverges and ak 3 bk then 2 ak diverges LimitComparison Test39 a If 11m b quot L gt0 then Zak and Zbk either both converge or both diverge k oo k a 39 If 11m quot 0 and Zbk converges then Zak also converges k gtoo bk a If 11m k 00 and Zbk diverges then Zak also diverges k xn bk THESE ARE TPIE TWO MOST FREQUENTLY USED TESTS IE SIZE COMPARISON IS EASY USE COMPARISON TEST OTHERWISE USE THE LIMITCOMPARISON TEST USUALLY USE PSERIES FOR THE COMPARISON SERIES ak1 ak Ratio Test Given Zak nd Iim oo If r lt1 then Zak converges If r gt1 then Zak diverges If r 1 the test is inconclusive USE FOR FACTORIALS AND POWERS OF K Root Test Given Zak nd lak r If r lt1 then Zak converges If r gt1 then Zak diverges If r 1 the test is inconclusive USE FOR POWERS OF R ESPECIALLY KK Alternating Series Test k1 The series 2 1 ak a1 a2 a3 a4 kl converges if all three of the following conditions are satis ed 1 The as are all positive r 2 3 1 ak1for all kgK for some integer K 3 elk gt0
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