MA122 - SERIES AND MULTIVARIABLE CALCULUS: TESTS FOR CONVERGENCE OF INFINITE SERIES

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1 MA22 - SERIES AND MULTIVARIABLE CALCULUS: TESTS FOR CONVERGENCE OF INFINITE SERIES ALVARO LOZANO-ROBLEDO Geometric Series Test, p-test, Divergece Test, Compariso Test, Limit Compariso Test, Itegral Test, Covergece of Absolute Value, the Ratio Test ad the Alteratig Series Test. As we kow, fidig out whether a series coverges or diverges ca be very difficult. I fact, there is o sigle trick that oe ca do to fid out. However, we have may good tricks at had, tests that ca tell us if the series coverges or diverges. Sometimes they succeed ad we fid out, or they fail ad we have to try a ew test. Test (Geometric Series Test). A geometric series =0 ar = a+ar+ar coverges a if ad oly if r <. If it coverges, its sum equals () Whe to use it. This test ca oly be used with geometric series. (2) How to use it. Check if the series is geometric: for example, divide cosecutive terms a 2 /a, a 3 /a 2,..., if the those quotiets are equal the the series is geometric ad r equals ay of the quotiets (r = a 2 /a for example). a is the first term of the series. If the series is geometric ad r < the the series coverges. Otherwise it diverges. Test 2 (The p-test). A series of the form coverges if p > ad diverges if p. p () Whe to use it. This test ca oly be used with series of the form / p. (2) How to use it. Simple, brig the series to the form / p. If p > it coverges. If p it diverges. r. Example The series diverges because: = ad /2 <. O the other had, the series /2 = coverges because 3/2 >. 3/2 Test 3 (The Divergece Test). Suppose a = a + a 2 + a 3 + a is a series ad a 0, the the series diverges. () Whe to use it. This test ca oly be used to prove divergece. We use this test whe the terms seem rather large, or whe we have the ituitio that a might ot be 0.

2 2 ALVARO LOZANO-ROBLEDO (2) How to use it. Calculate a. If the result is NOT zero the the series diverges ad we are doe (ad we wi). If the result if zero the test is icoclusive. No luck. Next test. Example Study the followig series for covergece: Notice that: = 3 0 Therefore, by the Divergece Test, the series diverges. Test 4 (Compariso Test). Suppose that 0 a b for all. () If b coverges, the a coverges. (2) If a diverges, the b diverges. () Whe to use it. We use this method whe the series remids us of aother series (usually oe like / p for some p) which we kow that coverges/diverges. The try to boud the give series by the series you kow. The Limit Compariso Test is usually easier to use! (see below). (2) How to use it. You are give a series. Your task is to boud the geeral term of the give series by aother expressio which you kow coverges (the you wat to boud above) or diverges (boud below). Example Does the followig series coverge? The series remids me of /2 which coverges (use p-test, for example). Next I try to boud the give series above by / 2. I fact: < 2 for all. Therefore, by the Compariso Test (with a = /( ) ad b = / 2 ), the series coverges. Test 5 (Limit Compariso Test). Let =0 a ad =0 b be two series of positive umbers. If the it a = L b

3 MA22 - SERIES AND MULTIVARIABLE CALCULUS:TESTS FOR CONVERGENCE OF INFINITE SERIES3 exists ad L 0 is a o-zero fiite umber, the both series =0 a ad =0 b coverge or both diverge. () Whe to use it. We use this method whe the series remids us of aother series (usually oe like / p for some p) which we kow that coverges/diverges. The Limit Compariso Test is usually easier to use tha the regular Compariso Test. (2) How to use it. You are give a series a which remids you of aother series b which you kow coverges (or diverges). Calculate: a = L b If L is ot zero ad ot ifiite the a coverges if ad oly if b coverges. If L = 0 or L = the the test is icoclusive. Tough luck. Maybe you did ot choose b correctly. Maybe it is time to move o to aother test. Example Does the followig series coverge? The series remids me of /2 which coverges (use p-test, for example). Next we compute the it: = Therefore, sice 0, by the Limit Compariso Test (with a b = / 2 ), the series coverges. = /( ) ad Example Does the followig series coverge? The series is a little bit too complicated to fid a similar ad easier series right away. However, if we forget about the lower order terms of : = 4 ad / is the harmoic series ad diverges by the p-test. Thus, we take b = / ad compute: ( ) Therefore the series diverges like the harmoic does / 2 + / 3 + / 3 + / = 4

4 4 ALVARO LOZANO-ROBLEDO Test 6 (Itegral Test - this test is ot required i this course). Suppose i= a i is a series with a i = f(i), where f(x) is a decreasig positive fuctio. The: If f(x)dx is fiite, the a i coverges. If f(x)dx is ifiite, the a i diverges. () Whe to use it. We use this method whe the geeral term i the series is a fuctio that we kow how to itegrate (easily). (2) How to use it. Calculate: f(x)dx = L If L is ot ifiite the a coverges. If L = the the series diverges. Example Does the series 2e 2 coverge? We calculate: [ 2xe x2 dx e x2] e e 2 = e Therefore the series coverges Tests for Series with positive ad egative terms. Test 7 (The Ratio Test). For a series a, suppose that the sequece of ratios a + / a has a it: a + = L. a If L < the a coverges. If L > or L = the a diverges. If L = the test tells us othig about the series a. () Whe to use it. This method is very useful whe the geeral term cotais powers (like 2 ) or factorials (!). (2) How to use it. Calculate: a + = L a ad follow the list above. Example Does the series followig series coverge? 2!

5 MA22 - SERIES AND MULTIVARIABLE CALCULUS:TESTS FOR CONVERGENCE OF INFINITE SERIES5 Here a = 2 /(!), thus a + = 2 + /(( + )!). We calculate the it: a + a 2 + (+)! 2! 2 +! 2 ( + )! 2 + = 0 Always have i mid that ( + )! = ( + )!. Sice 0 <, the series coverges. Test 8 (Test of covergece i absolute value). Suppose a is a series ad a coverges. The a coverges as well. () Whe to use it. We use this method for series with some egative terms, ad such that whe we forget the sig of the terms we recogize the series. (2) How to use it. Calculate a ad check a for covergece. If a coverges the a coverges. If a diverges the the test is icoclusive. Try the ratio test or the alteratig series test. Example Does the series ( ) coverge? Here a 2 = ( ) so a 2 = / 2. coverges as Moreover we kow that /2 coverges, by the p-test. Thus, well. Example Does the series ( ) coverge? Here a = ( ) so a = /. Moreover we kow that / diverges, by the p-test. Thus, the test is icoclusive ad we ca t coclude either way. ( ) 2 Test 9 (The Alteratig Series Test). Suppose {a } is a sequece of positive terms (a > 0), with: 0 < a + < a, i.e. the sequece is decreasig. a = 0. The the series ( ) a (or ( )+ a ) coverges. () Whe to use it. Wheever you wat to study a series of the form ( ) a, i.e. the sig of the geeral term is alteratig. (2) How to use it. Check that a > 0, a = 0 ad a + < a. If the three are true, the the series coverges. If ay of them fails, the test is icoclusive. Example Does the series followig series coverge? ( ) Here a = / > 0, thus a + = /( + ). First: + <

6 6 ALVARO LOZANO-ROBLEDO because + > for all. We calculate the it / = 0 ad we ca coclude that the series coverges. address: alozao@colby.edu Departmet of Mathematics (Mudd 406), Colby College, Waterville, Maie, 0490, U.S.A.