9.3 The INTEGRAL TEST; p-series
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1 Lecture 9.3 & 9.4 Math 0B Nguye of 6 Istructor s Versio 9.3 The INTEGRAL TEST; p-series I this ad the followig sectio, you will study several covergece tests that apply to series with positive terms. Note : Whe usig the Itegral Test, it s NOT ecessary to start at =. For istace, whe testig the series, we d use the improper itegral ( 3) 4 4 x 3 Example : Discuss the covergece / divergece of the series Solutio: dx.
2 Lecture 9.3 & 9.4 Math 0B Nguye of 6 Istructor s Versio Note : The series does t always coverge to the same value as that of the improper itegral. Note that the Itegral Test does t state the limit value to which the series coverges!!! Note 3: It s also NOT ecessary that fuctio f be always decreasig. What s importat is that f be ULTIMATELY decreasig [i.e., f is decreasig for x N for some costat N]. Example : Solutio: Test the series l for covergece / divergece.
3 Lecture 9.3 & 9.4 Math 0B Nguye 3 of 6 Istructor s Versio p-series Def: A series of the form positive costat.... is called a p-series, where p is a p p p p 3 For p =, the series... is called the harmoic series. 3 A geeral harmoic series is of the form a b The p-series has a simple arithmetic test for covergece or divergece. So, by the defiitio of the harmoic series, it s diverget. Example 3: Discuss the covergece / divergece of each series. a) b)
4 Lecture 9.3 & 9.4 Math 0B Nguye 4 of 6 Istructor s Versio 9.4 COMPARISONS of SERIES The two tests you study i this sectio allow you to compare oe series havig complicated terms with a simpler series whose covergece or divergece is kow (or ca be determied quite easily!!!). or, at least, let 0 a b for all N, for some N > 0 sice the covergece of a series is ot depedet o its first several terms. Note : Remember that both parts of the Direct Compariso Test require 0 a b, all positive terms. Iformally, this test says the followig about the series with positive terms:. If the larger series coverges, the smaller series must also coverge.. If the smaller series diverges, the larger series must also diverge. Note : Whe choosig a series for compariso, you may disregard all but the domiat term / the Highest Powers of i both the umerator ad deomiator. Example : Discuss the covergece / divergece of the series 4 Solutio: For large, the domiat term i the deomiator is, so we compare the give series with the series. We kow that is a coverget p-series with p = >. Also, term-by-term compariso yields 4 has a larger deomiator). for all (because the left side always By the Direct Compariso Test part, (sice the series coverges), the origial series must coverge. 4
5 Lecture 9.3 & 9.4 Math 0B Nguye of 6 Istructor s Versio Example : Determie the covergece / divergece of Solutio: The give series resembles (which is a diverget p-series with p ). But for all, does ot meet the requiremet of divergece (part of Direct Compariso Test). So, try a differet series: the harmoic series (which is also diverget!). Ad, term-by-term compariso yields for all > 4. (Do you see?) By the Direct Compariso Test part, diverges. Also read Examples & i textbook, pg. 6. Now, a give series may closely resemble a p-series or a geometric series, yet you caot establish the term-by-term compariso required to apply the Direct Compariso Test. Uder these circumstaces, you may be able to apply a secod compariso test called the Limit Compariso Test. Note 3: Remember that the limit value L must be fiite ad positive i order for you to apply the Limit Compariso Test ad have the correct coclusio. If L = 0, the you eed to chage the compariso series b.
6 Lecture 9.3 & 9.4 Math 0B Nguye 6 of 6 Istructor s Versio Example 3: Test the series Solutio: The: Sice Let the give series be for covergece or divergece. a lim lim lim > 0 b a. The choose the compariso series as is a coverget geometric series with coverget by the Limit Compariso Test.. r, the give series is Example 4: Determie the covergece or divergece of 3 4 Solutio: A reasoable compariso would be with the series, or 3. The let s fid out whether the compariso series is coverget or diverget. It is either a geometric series or a p-series, but: ' ' (l ) (l ) lim lim L H L H b lim lim 0 so by the th-term Test for Divergece, the series is diverget. Now: 3 a lim lim lim 3 3 b By the Limit Compariso Test, the give series diverges. 3 4 Also read Examples 3 & 4 i textbook, pgs
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