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Calculus II

by: Edgar Jacobi

Calculus II MATH 122

Edgar Jacobi
GPA 3.6

Charles Lamb Jr

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Charles Lamb Jr
Class Notes
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This 3 page Class Notes was uploaded by Edgar Jacobi on Monday September 7, 2015. The Class Notes belongs to MATH 122 at Kansas taught by Charles Lamb Jr in Fall. Since its upload, it has received 31 views. For similar materials see /class/182376/math-122-kansas in Mathematics (M) at Kansas.

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Date Created: 09/07/15
Series 00 ll Geometric series 2 T 711 00 1 2i Pserles 2177 n oo 3 Harmonic series 2 if Note this is a pseries with pl 711 n 00 W71 4 Alternating Harmonic series 2 n1 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 0 711 400 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 arn l 1 a 177 n 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 i1 00 n E ai m 3 E ail A question of central importance is how good close is your estimation 37 In the i1 139 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 ngtoo ngtoo n1 2 Integral Test Suppose fis a continuous positive decreasing function on N 00 Where N is a positive integer Then 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 ngtoo 12 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 lag L lt 1 then the series 221 an is absolutely convergent naoo quot b If l aquot1 l L gt 1 then the series 221 an is divergent a 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 00 a El Marl the antiderivative of 121 is arctanz n 00 b E ne integrate 16 by parts n1 F 5 533 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 El 51 n n 3quot4 2n oo 00 b 2 compare with geometric series 2 711 711 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 a n1 0 1 aga1nst E 27 711 n5 oo oo 00 b 21W against 217 217 77 71 n 1 1 Ema c sin 7 a ainst 7 usin lim 7 1 ltgt n n g gm 9 Alternating Series TestNote First test you can apply to apply to series with alternating termsi Previous three tests require positive terms 71 a n1 Ratio Test Note This test works for both types of terms of a series positive and alternating Use this contains factorlals ml and constants raised to the nth power 37h You want cancellation in the expression M i Also remember lim 1 er ngtoo m m Q r m 5 D S an A 3 M8 2 MSE EF b n 1


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