NTH, GEOMETRIC, AND TELESCOPING TEST

Transcription

1 NTH, GEOMETRIC, AND TELESCOPING TEST Sectio 9. Calculus BC AP/Dual, Revised 08 /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test

2 SUMMARY OF TESTS FOR SERIES Lookig at the first few terms of the sequece of partial sums may ot help us much 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 (expoetials or factorials)? Root Test: Does the series cotai a radical? Telescopig series: Will all but a couple of the terms i the series cacel out? Itegral Test: Ca you easily itegrate the expressio 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 would be better to use: Direct or Limit Compariso Test? /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test

3 DEFINITIONS A. Series is the sum of the terms i a sequece. B. Fiite sequeces ad series have defied first ad last terms, whereas ifiite sequeces ad series cotiue idefiitely. C. A series ca be writte with summatio symbol sigma, σ, the Greek letter S. D. S is ofte called a th partial sum, sice it ca represet the sum of a certai part of a sequece. E. Ifiite series coverges if the sequece of partial sums coverges to some umber, S. F. If the sequece of partial sums diverges, the the series diverges. /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 3

4 WHAT IS SIGMA NOTATION? Upper Limit Lower Limit Kow as idex Summatio Fuctio "The summatio from to of + ": /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 4

5 EXAMPLE Give the series σ = +, fid the first five terms of the sequece of partial sums, ad list them below. The, evaluate. Is the sequece of partial sums has a limit or boud? 3, 6,,8, /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 5

6 GEOMETRIC SERIES TEST A. Geometric series is i the form, σ = a (r) ; a 0 σ =0. a is the Iitial term of the series. r is the commo ratio B. Covergece vs. Divergece a (r) or. The geometric series coverges if r < to the sum of S = a r. The geometric series diverges if r /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 6

7 EXAMPLE Determie whether the followig series coverge or diverge, σ = it coverges, idetify the sum If a r 3 r /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 7

8 EXAMPLE Determie whether the followig series coverge or diverge, σ = it coverges, idetify the sum. 3 S r a S r Series is Coverget by GST; Sum is 3 3. If /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 8

9 EXAMPLE 3 Determie whether the followig series coverge or diverge, σ = 3 r 3 r 3 Series is Diverget by GST /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 9

10 EXAMPLE 4 Determie whether the followig series coverge or diverge, σ = /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test

11 EXAMPLE 4 Determie whether the followig series coverge or diverge, σ = a S r 3 5 S Series is Coverget by GST; sum is 3+ 5 /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test

12 YOUR TURN Determie whether the followig series coverge or diverge, σ = 3. If it coverges, idetify the sum. r a S r 3 S 4 Series is Coverget by GST; Sum is /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test

13 NTH TERM TEST FOR DIVERGENCE A. Oe Way Test (Divergece Oly) B. If lim a 0, the the series σ = a diverges C. Therefore, if lim a = 0, the the series DOES NOT coverge D. If the test does ot pass, the test is INCONCLUSIVE ad aother test must be used E. Use this test FIRST before others, due to time costraits /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 3

14 EXAMPLE 5 Determie whether the followig series coverge or diverge, σ = If it coverges, idetify the sum. 3 lim 3 5 lim Series Diverges by the th term Test /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 4

15 EXAMPLE 6 Determie whether the followig series coverge or diverge, σ =!!+. If it coverges, idetify the sum.! lim! x lim x x lim x 0 Series Diverges by the th term Test /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 5

16 EXAMPLE 7 Determie whether the followig series coverge or diverge, σ =.. lim. /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 6 S B 0 lim 0 a S r 0

17 EXAMPLE 7 Determie whether the followig series coverge or diverge, σ =.. 00 S Series Coverges by the GST; Sum is /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 7

18 YOUR TURN Determie whether the followig series coverge or diverge, σ = If it coverges, idetify the sum. 3 lim 3 3 lim lim Series Diverges by the th term Test /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 8

19 TELESCOPING TEST A. Kow as the terms collapse to oe term or several terms B. Uses the associative property of additio C. Series collapses to a fiite sum D. To get the sum, start pluggig i umbers E. Hit: Geerally has two ratios associated with Telescopig Series whe geeratig the sum /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 9

20 EXAMPLE 8 Determie whether the followig series coverge or diverge, σ = + +3 Start Pluggig i Numbers for. If it coverges, idetify the sum Series Coverges by the Telescopig Test; Sum is 3 /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 0

21 EXAMPLE 9 Determie whether the followig series coverge or diverge, σ = If it coverges, idetify the sum a b 3 a 3 b /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test

22 EXAMPLE 9 Determie whether the followig series coverge or diverge, σ = If it coverges, idetify the sum. a 3 b, a 3 a a 3, b 3 b b /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test

23 EXAMPLE 9 Determie whether the followig series coverge or diverge, σ = If it coverges, idetify the sum /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 3

24 EXAMPLE 9 Determie whether the followig series coverge or diverge, σ = If it coverges, idetify the sum Series Coverges by the Telescopig Test; Sum is /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 4

25 YOUR TURN Determie whether the followig series coverge or diverge, σ = +. If it coverges, idetify the sum. Series Coverges by the Telescopig Test; Sum is /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 5

26 AP MULTIPLE CHOICE PRACTICE QUESTION What is the value of (A) (B) 4 (C) 6 = (D) The series diverges (NON-CALCULATOR) + 3? /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 6

27 AP MULTIPLE CHOICE PRACTICE QUESTION What is the value of = (NON-CALCULATOR) + 3? Vocabulary Coectios ad Process Aswer Geometric Series k0 k0 3 3 S B /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 7 k0 3 k03 r 3 a r 3 3

28 ASSIGNMENT Page odd, 5-3 odd, 4-53 odd /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test 8