P-SERIES AND INTEGRAL TEST
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1 P-SERIES AND INTEGRAL TEST Sectio 9.3 Calcls BC AP/Dal, Revised 08 /4/08 0:8 PM 9.3: p-series ad Itegral Test
2 SUMMARY OF TESTS FOR SERIES Lookig at the first few terms of the seqece of partial sms may ot help s mch so we will lear the followig te tests to determie covergece or divergece: P A R R T I N G C p-series: Is the series i the form P? Alteratig series: Does the series alterate? If it does, are the terms gettig smaller, ad is the th term 0? Ratio Test: Does the series cotai thigs that grow very large as icreases (epoetials or factorials)? Root Test: Does the series cotai a radical? Telescopig series: Will all bt a cople of the terms i the series cacel ot? Itegral Test: Ca yo easily itegrate the epressio that defie the series? th Term divergece Test: Is the th term somethig other tha zero? Geometric series: Is the series of the form, σ =0 a r Compariso Tests: Is the series almost aother kid of series (e.g. p-series or geometric)? Which wold be better to se: Direct or Limit Compariso Test? /4/08 0:8 PM 9.3: p-series ad Itegral Test
3 P-SERIES TEST A. P i p-series rle is the epoet i the epoetial rle B. p-series is a series i the form of = = p p p 3 p p + where p is a positive costat C. The series will coverge if p > D. The series will diverge if p < Less Tha, Diverges /4/08 0:8 PM 9.3: p-series ad Itegral Test 3
4 EXAMPLE Use the p-series test to prove whether the series coverges or diverges, = 3 5/3. 3 5/3 p 5 5 ; p= ; 3 Series Coverges by the p-series Test /4/08 0:8 PM 9.3: p-series ad Itegral Test 4
5 EXAMPLE Use the p-series test to prove whether the series coverges or diverges, = /5 3 p ; 5 5 p= ; 5 Series Diverges by the p-series Test /4/08 0:8 PM 9.3: p-series ad Itegral Test 5
6 EXAMPLE 3 Use the p-series test to prove whether the series coverges or diverges, = p= ; Series Coverges by the p-series Test /4/08 0:8 PM 9.3: p-series ad Itegral Test 6
7 YOUR TURN Use the p-series test to prove whether the series coverges or diverges, =. 3 p= >; Series Coverges by the p-series Test /4/08 0:8 PM 9.3: p-series ad Itegral Test 7
8 INTEGRAL TEST A. Where f for ad a = f where σ = f ad f d : Remember PCD for Itegral Test ( PEOPLE COUNTING DOWN ). Positive: if graphed, above the -ais. Cotios: Use defiitio of polyomials or are cotios i their domai like e or l 3. Decreasig; takig the derivative 4. Aswer as: By the itegral test, sice f d coverges/diverges, the f coverges/diverges. σ = 5. I other words, as the improper itegral behaves, so does the series, ad vice versa. 6. This DOES NOT mea that the series coverges to the vale of the defiite itegral 7. The fctio eed oly be decreasig for all > k for some k. /4/08 0:8 PM 9.3: p-series ad Itegral Test 8
9 EXAMPLE 4 Use the Itegral Test to prove whether the series coverges or diverges,. = POSITIVE? 3+5 f 3 5 CONTINUOUS? DECREASING? Positive ad Cotios f f 5 f 3 f f f /4/08 0:8 PM 9.3: p-series ad Itegral Test 9
10 EXAMPLE 4 Use the Itegral Test to prove whether the series coverges or diverges, = d 3 5 d lim b b 3 5 b lim 3 b /4/08 0:8 PM 9.3: p-series ad Itegral Test 0 8 d d 35 d 3d d d 3 8
11 EXAMPLE 4 Use the Itegral Test to prove whether the series coverges or diverges,. = b d lim d 3d b 3 8 f lim l b 3 /4/08 0:8 PM 9.3: p-series ad Itegral Test lim l l 8 b 3 b 8 is positive, cotios, ad decreasig o I,. Sice d diverges, diverges by the Itegral Test d d 3 8
12 EXAMPLE 5 Use the Itegral Test to prove whether the series coverges or diverges, = +. f Positive ad Cotios 5 5 f f f POSITIVE? CONTINUOUS? DECREASING? /4/08 0:8 PM 9.3: p-series ad Itegral Test f 5 f 3 f
13 EXAMPLE 5 Use the Itegral Test to prove whether the series coverges or diverges,. = + d lim b b /4/08 0:8 PM 9.3: p-series ad Itegral Test 3 b lim b lim b d l d b d d d d
14 EXAMPLE 5 Use the Itegral Test to prove whether the series coverges or diverges,. b = + lim l b lim l l b f lim l b is positive, cotios, ad decreasig o I,. Sice d diverges, diverges by the Itegral Test. /4/08 0:8 PM 9.3: p-series ad Itegral Test 4
15 EXAMPLE 6 Use the Itegral Test to prove whether the series coverges or diverges,. = POSITIVE? + f Positive ad Cotios f f 5 f 3 f f f CONTINUOUS? DECREASING? /4/08 0:8 PM 9.3: p-series ad Itegral Test 5
16 EXAMPLE 6 Use the Itegral Test to prove whether the series coverges or diverges,. = + d lim b b lim b b /4/08 0:8 PM 9.3: p-series ad Itegral Test 6 a lim arcta b d d b d d
17 EXAMPLE 6 Use the Itegral Test to prove whether the series coverges or diverges, = +. f lim arcta b lim arcta arcta b /4/08 0:8 PM 9.3: p-series ad Itegral Test is positive, cotios, ad decreasig o I,. Sice d coverges, coverges by the Itegral Test. DO NOT WRITE: Series Coverges to b 4 by the Itegral Test a, a, d d
18 YOUR TURN Use the Itegral Test to prove whether the series coverges or diverges, = 4 +. f 5 5 f Positive ad Cotios f f 4 POSITIVE? CONTINUOUS? DECREASING? /4/08 0:8 PM 9.3: p-series ad Itegral Test 8 f 5 f 3 f
19 YOUR TURN Use the Itegral Test to prove whether the series coverges or diverges,. = d 4 a b lim d a b 4 d d lim b b b lim b d a d a d d /4/08 0:8 PM 9.3: p-series ad Itegral Test 9
20 YOUR TURN Use the Itegral Test to prove whether the series coverges or diverges,. b d = 4 a + lim b a a b d lim arcta d b lim arcta arcta b f lim b 4 8 I is positive, cotios, ad decreasig o,. Sice d coverges, coverges by the Itegral Test. 4 4 /4/08 0:8 PM 9.3: p-series ad Itegral Test 0 4 d d
21 SPECIAL CASE: HARMONIC SERIES A. For p =, = = 3 harmoic series ad diverges + the series is called the B. The Harmoic Series diverges becase eve thogh the sm approaches to zero, it is too slow to coverge ad therefore, diverges. /4/08 0:8 PM 9.3: p-series ad Itegral Test
22 /4/08 0:8 PM 9.3: p-series ad Itegral Test
23 VISUAL CASE Give = y = / y Itegral is DECREASING by sig the LEFT RIEMANN SUM, therefore,, 3, 3 4, 4 /4/08 0:8 PM 9.3: p-series ad Itegral Test 3 p Series : DIVERGENT d Sice the covergece to zero is so slow, therefore this eqatio is diverget
24 PROOF Use the Itegral Test to prove whether the series coverges or diverges,. = f Positive ad Cotios f 5 POSITIVE? CONTINUOUS? DECREASING? /4/08 0:8 PM 9.3: p-series ad Itegral Test 4 5 f 3 3 f f 5 f 3 f 5 3
25 PROOF Use the Itegral Test to prove whether the series coverges or diverges,. = d lim b d b lim b b d lim l b b /4/08 0:8 PM 9.3: p-series ad Itegral Test 5
26 PROOF Use the Itegral Test to prove whether the series coverges or diverges,. = lim l b b lim l l b f is positive, cotios, ad decreasig o I,. Sice d diverges, diverges by the Itegral Test. /4/08 0:8 PM 9.3: p-series ad Itegral Test 6
27 EXAMPLE 7 Use the Itegral Test to prove whether the series coverges or diverges,. =0,000 POSITIVE? f Positive ad Cotios f 5 5 f 3 3 f CONTINUOUS? DECREASING? f 5 f 3 f 5 3 /4/08 0:8 PM 9.3: p-series ad Itegral Test 7
28 EXAMPLE 7 Use the Itegral Test to prove whether the series coverges or diverges, =0, lim b b0000 lim b d b0000 lim l b d /4/08 0:8 PM 9.3: p-series ad Itegral Test 8 d b 0000
29 EXAMPLE 7 Use the Itegral Test to prove whether the series coverges or diverges, =0,000. f lim l b b 0000 lim l l 0000 b is positive, cotios, ad decreasig o 0000,. Sice d diverges, diverges by the Itegral Test /4/08 0:8 PM 9.3: p-series ad Itegral Test 9
30 EXAMPLE 8 Use the Itegral Test to prove whether the series coverges or diverges, =0,000. f Positive ad Cotios f f f POSITIVE? CONTINUOUS? DECREASING? f 5 f 3 f 5 9 /4/08 0:8 PM 9.3: p-series ad Itegral Test 30
31 EXAMPLE 8 Use the Itegral Test to prove whether the series coverges or diverges, =0, lim b b0000 lim b b 0000 lim b d /4/08 0:8 PM 9.3: p-series ad Itegral Test 3 d d b 0000
32 EXAMPLE 8 Use the Itegral Test to prove whether the series coverges or diverges, =0,000. lim b b 0000 lim b f is positive, cotios, ad decreasig o 0000,. Sice d coverges, coverges by the Itegral Test /4/08 0:8 PM 9.3: p-series ad Itegral Test 3
33 AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR) Which of the followig series is/are coverget? I II (A) Noe. (B) I oly. (C) II oly. (D) I ad II. /4/08 0:8 PM 9.3: p-series ad Itegral Test 33
34 AP MULTIPLE CHOICE PRACTICE QUESTION (NON-CALCULATOR) Which of the followig series is/are coverget? Vocablary Coectios ad Process Aswer p-series Coverget I ; p series : 3 Coverget II ; p series : 3 Harmoic Series; Diverget B /4/08 0:8 PM 9.3: p-series ad Itegral Test 34
35 ASSIGNMENT Page odd, 5-37 odd, 43 /4/08 0:8 PM 9.3: p-series ad Itegral Test 35
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