3.1 & 3.2 SEQUENCES. Definition 3.1: A sequence is a function whose domain is the positive integers (=Z ++ )

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1 3. & 3. SEQUENCES Defiitio 3.: A sequece is a fuctio whose domai is the positive itegers (=Z ++ ) Examples:. f() = for Z ++ or, 4, 6, 8, 0,. a = +/ or, ½, / 3, ¼, 3. b = /² or, ¼, / 9, 4. c = ( ) + or 0,, 0, Defiitio 3.: A sequece is said to have the limit L, if for ay є > 0, however small, there is some value N, such that a L < є wheever > N. Such a sequece is said to be coverget, ad we write its limit as lim a = L. Example: a = +/ ½ / 3 ¼ L= L = lim a = because for ay є > 0 there exists N (e.g. N=/є+½) such that a < є wheever > N. N }є }є

2 Defiitio 3.: If a sequece has o limit, it is diverget. Example: c = ( ) + c is diverget because for ay L ad for some є > 0 there does exists N such that c L < є wheever > N. L N }є }є Defiitio 3.: If a sequece is said to be defiitely diverget if either oe of the followig coditios holds: (i) If for ay (arbitrary large) value of K there is a N sufficietly large that a > K for > N. I this case we write lim a =. (ii) If for ay (arbitrary large) value of K there is a N sufficietly large that a < K for > N. I this case we write lim a =. Example: Sequece d() = is defiitely diverget ad lim d = Because as for ay K, there is N = K/ (for example) such that d = > N = K for ay > N.

3 3.3 PRESENT VALUE CALCULATIONS Preset value (PV) of a future paymet (V(t)) ca be computed as PV = V( t ) ( + r) t where t stads for time ad r for a et iterest rate. I other words future paymet is discouted by a discout factor (or discout rate) /( + r) for each period of waitig. Usig the same equatio, we ca also see how the future paymet (its omial value) icreases i time: V( t ) = PV ( + r) t EXAMPLE: Determie how much moey a ivestmet of $0,000 will geerate at the ed of oe year give the aual iterest rate is 0%. Assume the iterest paymet is paid oce. ( ) t ( ) V( ) = PV + r = $ 0, % = $, 000 3

4 Now assume that iterest is paid twice a year (=semiaual compoudig): t r V( ) PV = + = $ 0, 000( + 5% ) = $ 0, 500( + 5% ) = $, 05 Now assume that iterest is paid mothly (=mothly compoudig): t r 0% V( ) = PV + = $ 0, = $, Now assume that iterest is paid daily (=daily compoudig): 365t 365 r 0% V( ) = PV + = $ 0, = $, I 683 Jakob Berolli, a so of a Swiss baker, asked what happes if iterest is compouded cotiuously 4

5 The relevat calculatio ca be writte as: st r V( t ) = lims PV + = lims PV + = s s r rt s rt r = lims PV + PV lim s = + r where =s/r the ever icreasig umber of periods withi year/r. As, the time is thus sliced cotiuously. Beroulli oticed that lim + st is betwee ad 3, but the limit was oly solved by aother so of a aother Swiss baker Leohard Euler i 77. He coveietly called the ew costat e (as a costat a was already i use): lim + = e

6 Because V( t ) = PV lim + thus the relevat calculatio is rt rt V( t ) = PVe ad i our example st r rt 0% = s + = = = V( ) PV lim $, e $, e $,. s Exercises: The uiverse is curretly gazzilio miles wide. It grows cotiuously at a rate % per year. How big it will be i 00 years? V( ) PVe e e. 00 = rt = %* = 7 It will be about.7 gazzilio miles wide. 6

7 Your ivestmets required immediate costs of $00. It will geerate oe time reveue of $400 at the ed of the first year. What is its aual rate of retur? (a) if the iterest is compouded aually? i = 300% 400 = 00 + i i = 3 ( ) (b) if the iterest is compouded twice a year? i 00% 400 i + = 00 i = ( ) (c) if the iterest is compouded cotiuously? r = l(4) 39% ( r ) t V( t ) = PV e = PVe 400 = 00e r r = l ( 4 ) rt 7

8 Your ivestmets required immediate costs of $00. It will geerate oe time reveue of $400 at the ed of the secod year. What is its aual rate of retur? (a) if the iterest is compouded aually? i = 00% 400 = 00 + i i = ( ) (b) if the iterest is compouded twice a year? i 83% 400 i + 4 = 00 i = ( ) (c) if the iterest is compouded cotiuously? r = l(4) / 69% ( r ) t V( t ) = PV e = PVe e r r = l 4 = ( ) rt 8

9 Suppose that the cost of stream of paymets $0, $70, $60 is $00. First paymet of 0$ is paid oe year after the purchase, the secod paymet of $70 is paid two years after the purchase ad the fial paymet of $60 is paid 3 years after the purchase. What is its aual rate of retur? (a) if the iterest is compouded aually? i = 0 % Note that the above are the same. (b) if the iterest is compouded cotiuously? r = l(.) 8% 0 = 00 e r r = l (.) 9

10 3.4 PROPERTIES OF SEQUENCES THEOREM 3. : Suppose that a ad b are coverget sequeces with limits L a ad L b respectively. It follows that: () lim + c a = c L a for c () lim + ( a ± b ) = L a ± L b (3) lim + ( a )( b ) = L a L b (4) lim + ( a / b ) = L a / L b if L b 0 THEOREM 3. : Suppose that a is a coverget sequece with limit L a ad a is a defiitely diverget sequece with limit +, ad c is a costat. It follows that: () lim + c b = + for c > 0, ad for c < 0 () lim + ( a + b ) = + (3) lim + ( a b ) = (4) lim + ( a )( b ) = + for L a > 0, ad for L a < 0 (5) lim + ( a / b ) = 0 ad as a special case of (5), lim oo ( c / b ) = 0 for c. 0

11 Use the results that lim + = 0, lim + = + ad theorems 3. & 3. to show that + (i) lim + 0 = lim lim + + = 0 because of Theorem 3. (3) + = 0 because of Theorem 3. () (ii) lim lim lim = + ( ) = 0 because of Theorem 3. () ( ) + = + because of Theorem 3. ()

12 Defiitio 3.5: A sequece is mootoically icreasig if a < a 3 < a 3 < ad it is mootoically decreasig if a > a 3 > a 3 > I either case is said to be mootoic. Defiitio 3.5: A sequece is bouded if ad oly if it has a lower boud ad a upper boud. Decide if the followig sequeces are mootoic ad bouded:. d = mootoically icreasig o upper boud. a = +/ mootoically decreasig bouded 3. b = /² mootoically icreasig bouded 4. c = ( ) + ot mootoic bouded

13 THEOREM 3.3 : A mootoic sequece is coverget if ad oly if it is bouded. M ( C B ) or M ( C B ) Note that Theorem 3.3 oly applies to mootoic sequeces. M does ot imply aythig. Use theorem 3.3 to show that the sequece a = /, =,,3, is coverget. Mootoic (as a + =a /<a ) Bouded (by for example above ad 0 below) Use theorem 3.3 to show that the sequece a = -, =,,3, is diverget. Mootoic (as a + =a <a ) Not bouded (It is ot bouded below.) 3

14 3.5 SERIES Defiitio 3.7 : If, t =,,3,... is a sequece, the a t EXAMPLE: a s = t = s a t, =,,3,.. is called a SERIES. = {, 3, 5, 7... } = {, 4, 9,6... } Theorem 3.4: If s = = a t is the series associated with sequece a t ad it follows that: t lim a a + = (i) if L < the the sequece s coverges (ii) if L > the the sequece s diverges (iii) if L = the the sequece s may coverge or diverge. L 4

15 Show that Geometric Series, a series costructed from the sequece a = ax coverges if ad oly if x <. As lim a a ax + + = = x < ax (Theorem 3.4), this series coverges. Where does it coverge to? =. Note that A ax x 3 A ax ax ( + x + x + x +...) ad thus A = x. This ca be simplified to It coverges to A 3 = ax = ax ( + x + x + x +... ) = ax A = x. t Let ( 3 s = a = + a + a + a +...a ) t=. Solve for s. Whe does it coverge? 3 3 s a a a...a = = + a + a + a + a +...a a = a + as ( ) ( ) Thus s ( = a ) / ( a). If a limit is / ( a). < < (Theorem 3.4) the it coverges. Its 5

16 Note that Theorem 3.4 does ot help with all series. For example P-series, a series costructed from the sequece a = / p, have L=. It coverges if ad oly if p >. Example: Harmoic Series (p=): + / + /3 + /4 + /5 + /6 + / 7 + / 8 + /9 + / > + / + / + / + / +... Harmoic Series is mootoic ad ot bouded ad thus it diverges. Example: Over-harmoic Series with p=: + /4 + /9+/6 + /5+/36+/49+/64 + /8+ /6 + < + /4 + /8 + 4/3 + 8/8 + = + / + /4 + / 8 + / 6 + < This series is mootoic ad bouded ad thus it coverges. ( It coverges to π /6.) 6

17 Defiitio 3.8 : The iteral rate of retur of a project or ivestmet is the rate of iterest that equates the preset value of beefits ad costs. Suppose the cost of the stream of yearly paymets that starts i oe year is $0, $0x(.04), $0x(.04) x(.04), of a ivestmet is $60. What is the Iteral Rate of Retur (IRR) i? ( ) ( ) ax A = ( ) ( ) where x A = $ 60, a = $ 0 / ( + i) ad x =. 04 / ( + i ). + $ 0 $ $ $ 60 = i i i i 3( + i ) = = = + i + i. 04 i i ( i. ) = i = +. = ( 3 )% 60 = 0 ( 0 04) = ( 37 3 ) / i. i % 7

18 Suppose that the iitial start up costs ad clea up costs, paid 5 years later, are $00 each. The reveue is $00 i year. What is (are) ALL the Iteral Rates of Retur (IRR) i? I additio to the above, a egative -.5, ad complex umbers i ad i are also mathematically correct solutios. 8

19 Fially that IRR does ot take ito the accout the duratio of the project. Sometimes a smaller retur o larger ivestmet may be the better alterative: 9

20 EXAM () Logic () Chapter (sets, ) (3) Chapter 3 (sequeces, ) You ca brig: book calculator (o graphig) otes You caot brig: ANY HOMEWORK OR TEST 0

10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.

10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term. 0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece