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# Class Note for MATH 122 with Professor Lamb Jr at KU

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Kansas taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15

Series 00 ll Geometric series 2 T 711 oo 2 Pseries E nip 711 oo 3 Harmonic series 2 Note this is a pseries with pl 711 1 1 oo 4 Alternating Harmonic series 2 n 711 Theorems 1 8il Theorem 7 Every bounded monotonic sequence is convergenti oo 2 8i2 Theorem 6 If the series 2 an is convergent then lim an 0i n1 naoo oo 3 534 Theorem 1 IF a series 2 an is absolutely convergent then it is convergenti 711 4 A geometric series is convergent when M lt l and divergent when M 2 1 When M lt 1 it sum is 00 1 n7 7 a E m 7 4 711 oo 5 The pseries E nip is convergent if p gt 1 and divergent when p S 1 711 Estimation In many cases one cannot explicitly compute the exact sum of a series lnstead one can estimate the 00 exact sum 2 ai using a partial sum with only nitely many terms iiei n terms The estimation is 391 00 1 7L 2 ai m 3 E ail A question of central importance is how good close is your estimation 37 In the i1 i1 following two cases we have answers 1 533 Theorem 3 Error Estimate for lntegral Test Suppose Sun is convergent and let be the function used within the Integral test then the error in estimation Rn s 7 3n sati es the following inequality Af rm R mfrdr 2 534 Alternating Series Estimation Theorem If s Exilyklbn is the sum of the alternating series then the error in estimation satis es the following inequality anl ls 7 Snl S bn1 Tests 00 l Divergence Test If lim an does not exist or lim an 0 then the series 2 an is divergent naoo naoo n1 2 Integral Test Suppose fis a continuous positive decreasing function on N 00 Where N is a positive integer Then 00 a If is convergent then 2 an is convergent n1 b If is divergent then 2 an is divergent n1 3 Comparison Test Suppose that 2 an and E 12 are series With positive terms a If 21 is convergent and an S bn then Sun is convergent b If 21 is divergent and an 2 bn then Sun is divergent 4 Limit Comparison Test Suppose that 2a and 2b are series With positive terms If an hm i c naoo 7 Where c is nite and c gt 0 then either both series converge or both series diverge 5 Alternating Series Test If the alternating series 22171W 1bn Inn gt 0 satis es a 12711 S In for all n b lim 1 0 naoo then the series converges 6 Ratio Test a If lim L lt 1 then the series 221 an is absolutely convergent naoo quot b If lim L gt 1 then the series Emil an is divergent naoo aquot n c If lim L l the test is inconclusive use another test naoo n Examples 1 Divergence Test Note Always use this test rst If the series is divergent done Otherwise use another test 00 DO 2 1 2 1 2 2 1 a E 3127 an 322 A g E 312 1s d1vergent n1 n1 00 b 2 7271 an A 0 use another test to check for convergence n1 2 Integral Test Note Apply this test When you can integrate fx a E f the antiderivative of 711 1271 is arctanz 00 b E ne integrate 16 by parts n1 3 Comparison Test Note Need another series Ebn for comparison Two main series to use for com parison are geometric and pseriesi oo 00 a 21 5 compare with pseries E 51 n b 4 n1 711 oo 2 7 compare with geometric series 2 711 4 Limit Comparison Test Note Use this test when you would like to use the comparison test7 however the series against which you are comparing7 has terms that you can t make satisfy an S In or for divergence an 2 bn oo 1 oo 1 a E m aga1nst E 27 n1 n1 b n5 n71 m m n 7 1 1 1 Ema c E sin7 against E 7 using lim T 1 n1 n n1 n 9 H0 5 Alternating Series TestNote First test you can apply to apply to series with alternating termsi Previous three tests require positive terms a D18 1 1 n 1 6 Ratio Test Note This test works for both types of terms of a series positive and alternating Use this test when an contains factorials ml and constants raised to the nth power 37h You want cancellation in the expression Also remember lim 1 e quot HHOO A 3 M8 was S H A 55 S M8 ali

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