Math 20B. Lecture Examples.

Transcription

1 Math 20B. Leture Examples. (7/9/09) Setio 0.3. Covergee of series with positive terms Theorem (Covergee of series with positive terms) A ifiite series with positive terms either overges or diverges to. The series overges if its partial sums are bouded ad diverges if its partial sums are ot bouded. Theorem 2 (The Itegral Test for series with positive terms) Suppose that the series = 0 a () with positive terms is suh that a = f() for itegers with some, where y = f(x) is otiuous o [, ) ad dereasig for x. The the ifiite series () overges if ad oly if the improper itegral overges. f(x) dx (2) Figures ad 2 show why the series overges () if the itegral (2) overges. The area of the retagles i Figure is less tha the area of the regio i Figure 2. If the itegral overges, the the area of the regio i Figure 2 is bouded as N, so that the partial sums of the series are bouded ad the series overges. Area = a Area = N =+ FIGURE FIGURE 2 f(x) dx Leture otes to aompay Setio 0.3 of Calulus, Early Trasedetals by Rogawski.

2 Math 20B. Leture Examples. (7/9/09) Setio 0.3, p. 2 Figures 3 ad 4 show why the series () diverges if the itegral (2) diverges. The area of the retagles i Figure 4 is greater tha the area of the regio i Figure 3. If the itegral diverges, the the area of the regio i Figure 3 teds to as N, so that the partial sums of the series ted to ad the series diverges. N+ a = Area = f(x) dx Area = FIGURE 3 FIGURE 4 The Itegral Test is used to establish the followig result. Theorem 3 (Covergee of the p-series) The ifiite series overges if p > ad diverges if p. p = + 2 p + 3 p + 4 p + Example Example 2 Example 3 Aswer: with p =.75 >. overge or diverge ad why? = =2 9 overges beause it is a ostat multiplied by the p-series overge or diverge ad why? Aswer: 9 = diverges beause it is a ostat multiplied by the p-series 9 /2 /2 =2 =2 =2 with p = 2 <. 7 + Aswer: The series diverges beause a = overge or diverge ad why? 7 + = 7 + / 7 as ; the terms do ot ted to 0.

3 Setio 0.3, p. 3 Math 20B. Leture Examples. (7/9/09) Example 4 Aswer: The series is alled the harmoi series. it overge or diverge? diverges beause it is the p-series with p =. Theorem 4 (The Compariso Test with positive terms) Suppose that a ad b are series with positive terms. = 0 = 0 (a) If b overges ad there are ostats M ad N suh that a Mb for N, = 0 the a also overges. = 0 (b) If b diverges ad there are ostats M > 0 ad N suh that a Mb for N, = 0 the a also diverges. = 0 Example 5 the series =0 (0.) + overge or diverge ad why? (0.) Aswer: overges by the Compariso Test with the overget geometri series (0.) beause + =0 =0 (0.) + (0.) for 0. (The partial sums of the first series i Figure A7a are less tha the partial sums of the seod series i Figure A7b.) y y y = =0 N (0.) + Figure A4a 0 20 N y = (0.) =0 Figure A4b

4 Math 20B. Leture Examples. (7/9/09) Setio 0.3, p. 4 Example overge or diverge ad why? 2 + Aswer: 5 3 diverges by the Compariso Test with the diverget harmoi series ( ) > = for, ad is a positive ostat. 5 5 beause Theorem 5 (The Limit Compariso Test with positive terms) Suppose that a lim = L b where the a s ad b s are positive ad L is either a oegative umber or. (a) If L is fiite ad positive, the a Lb for large ad a overges if ad oly =j 0 if b overges. = 0 (b) If L = 0, the a is muh smaller tha b for large ad overges. () If L =, the a is muh larger tha b for large ad diverges. = 0 a overges if j= 0 a diverges if = 0 b = 0 b Example 7 Aswer: (Example 5 redoe) Use the Limit Compariso Test to determie whether (0.) the series overges or diverges. + (0.) =0 + (0.) = + L = 0 as (0.) overges by the Limit Compariso Test with + =0 the overget geometri series =0 (0.) beause L = 0.

5 Setio 0.3, p. 5 Math 20B. Leture Examples. (7/9/09) Example 8 Example 9 Example 0 (Example redoe) Use the Limit Compariso Test to determie whether overges or diverges. Aswer: = (2 + ) 5 3 = L = 5 as Compariso Test with the diverget harmoi series Aswer: beause 0(2 ) < L <. Aswer: Iterative Examples 0(2 ) 5 beause 0 < L <. overge or diverge ad why? (2 ) 5 overges by the Limit Compariso Test with the overget geometri series ( = 5 0(2 ) 2 5 the series ) = = 5 ( = 5 0(2 ) 2 5 ( /5 ) + 4 Compariso Test with the overget p-series ) overge or diverge ad why? = L = 5 as beause 0 < L <. 3 diverges by the Limit ( 2 5) =0 0 L = 0 as, ad /5 2 overges by the Limit + Work the followig Iterative Examples o Shek s web page, http// ashek/: Setio 0.3: Examples 4 Setio 0.4: Examples 4 The hapter ad setio umbers o Shek s web site refer to his alulus mausript ad ot to the hapters ad setios of the textbook for the ourse.