5.6 Absolute Convergence and The Ratio and Root Tests
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1 5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch 5.6 Absolute Covergece ad The Ratio ad Root Tests Recall from our previous sectio that diverged but ( ) coverged. Both of these sequeces have b =. Agai, oe coverges ad the other does ot. We itroduce two terms to distiguish betwee these two cases. Let a = a + a 2 + a 3 + a Defiitio 5. (Absolute Covergece). A series a is Absolutely Coverget if the series a is coverget. A quick ote: All coverget series with positive terms are automatically absolutely coverget sice a = a Cosider the followig two examples: Example ( ) 2 ( ) 2 = which coverges 2 So the series ( ) 2 is absolutely coverget. Example 5.5. ( ) Eve though ( ) coverges, ( ) = diverges 269
2 5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch So we caot say it s absolutely coverget. But the origial series did coverge. We ow itroduce our secod term for covergece. Defiitio 5.2 (Cotitioal Covergece). If a coverges but a diverges, the the series a is said to be Coditioally Coverget Theorem 5.6. If a series a is absolutely coverget, the a is coverget. Example Determie if cos() 2 coverges or diverges. The previous theorem states that if it s absolutely coverget, the it coverges. Let s check for absolute covergece. cos() 2 = cos() 2 2 By the Direct Compariso Test, sice coverges, the 2 cos() coverges 2 Therefore, cos() 2 coverges absolutely We ow go over our last two tests to determie covergece or divergece Ratio Test. If lim a + a = L <, the a is absolutely coverget. 2. If lim a + a = L >, the a is diverget. 270
3 5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch 3. If a + a =, the the test is icoclusive. You eed to use a differet test. Example Determie if ( ) 4 7 coverges. a + = ( )+ ( + ) 4, ad a 7 + = ( ) 4 7 a + a ( ) + ( + ) ( ) 4 ( + ) = 7 lim ( + ) 4 = 4 7 < Sice L = 7 <, the Ratio Test cocludes ( ) 4 is absolutely coverget Determie if! coverges. Sice lim = 0, we kow it diverges. But let s go ahead ad show it with the! Ratio Test. a + = ( + )+, ad a = ( + )!! a + a ( + ) + ( + )!! ( + ) ( + ) ( ) = lim ( + ) ( + = lim + ) 27! ( + )!
4 5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch This is a L Hospital Problem, which I ll let you work o. If you do it correctly, you should get ( + = e > ) Sice L = e >, the Ratio Test cocludes 3. Use the Ratio Test o ad 2.! diverges. We kow diverges ad coverges. Let s see what the Ratio Test tells us. 2 (a) a + = +, ad a = a + a + = The Ratio Test states that if L =, the test is icoclusive. We would eed to use aother test to determie its covergece. Good thig we have the p-series test, right? (b) 2 a + = ( + ) 2, ad a = 2 a + a ( + ) 2 2 = 272
5 5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch The Ratio Test states that if L =, the test is icoclusive. We would eed to use aother test to determie its covergece. 4. Determie if (2)! coverges. a + = (2( + ))!, ad a = (2)! (2( + ))! (2)! (2)! (2 + 2)! The trick here is to write out a few terms of the factorial util it matches up with aother factorial. (2 + )! = (2 + 2)(2 + )! = (2 + 2)(2 + )(2)! (2)! (2 + 2)! (2)! (2 + 2)(2 + )(2)! (2 + 2)(2 + ) = 0 Sice L = 0 <, the Ratio Test cocludes (2)! is absolutely coverget Root Test. If lim a = L <, the a is absolutely coverget. 2. If lim a = L >, the a is diverget. 273
6 5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch 3. If lim a = L =, the the test is icoclusive. You must use a differet test to determie covergece. The root test is useful whe you have a sequece raised to the -th power i some way, Example Determie if ( ) 6 3 coverges. + 4 ( a = (b ) ) = lim = 6 < Sice L = 6 <, the Root Test cocludes ( ) 6 3 is absolutely coverget Determie if ( + ) 2 coverges. ( + ) 2 = lim ( ( + ) ) ( = lim + ) You evaluate this limit usig L Hospitals Rule. If you do it correctly, you get ( lim + ) = e Sice L = e <, the Root Test cocludes ( + ) 2 coverges. 3. Is there ay value of k that makes 2 coverge? For example, does somethig like k 2 coverge? 00,000,000,
7 5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch Let s go ahead ad use the Ratio Test. a + = 2+ ( + ) k, ad a = 2 k 2 + ( + ) k k 2 2 k ( + ) k = 2 Sice L = 2 >, the Ratio Test cocludes 2 4. Does l coverge? diverges for all values of k. k Sice we re still i the Ratio ad Root Test sectio, we might as well use it. Sice it does t have a power of, we ll use the Ratio Test. a + = l( + ), ad a = l l( + ) l() l() l( + ) = lim LH + = lim + = Sice L =, the Ratio Test is icoclusive. We eed aother test. We ca do a Direct Compariso with. Sice l < for all, the l > for all. Sice diverges, by the Direct Compariso Test, so does l. 275
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