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# Note for MATH 250A at UA

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Date Created: 02/06/15

MATH 250a Fall Semester 2007 Section 2 J M Cushing Tuesday October 30 httpmathariz0naeducushing250ahtml Formal Definition A sequence x converges if there is a number L called the limit of the sequence for which the following is true for any small positive number 6 there is a number N so that xn Llt forall ngtN The sequence gets and remains as close to L as you want provided you go far enough out the sequence Example Prove that x 05 converges to L O Example Prove that x 05 converges to L 0 Given any 6 gt 0 it is true that x LI 2 05 lt 5 if 1n 0 5 lt 1n 5 Example Prove that x 05 converges to L 0 Given any 6 gt 0 it is true that x LO5 lt5 if 1r105 lt1r1 n1r105 lt1r1 Example Prove that x 05 converges to L 0 Given any 6 gt 0 it is true that x LI 2 05 lt5 if ngtlr1 1r105N Example Prove that x 05 converges to L 0 Given any 6 gt 0 it is true that x LI 2 05 lt5 if ngtlr1 1r105N For example x lt 01 provided n gt1r1011r105 332 Example Prove that x 05 converges to L 0 Given any 6 gt 0 it is true that x LI 2 05 lt5 if ngtlr1 1r105N For example x lt 01 provided n gt1r1011r105 332 x lt001 provided ngt1n0011n05664 Example Prove that x 05 converges to L 0 Given any 6 gt 0 it is true that x LI 2 05 lt5 if ngtlr1 1r105N For example x lt 01 provided n gt1r1011r105 332 x lt001 provided ngt1n0011n05664 Similarly we can show xn r converges to 0 if lrl lt 1 xn r diverges if lrl 21 Formal Definition A sequence xn converges if there is a number L called the limit of the sequence for which the following is true for any small positive number 6 there is a number N so that xn Llt forall ngtN One dif culty in using this de nition to calculate limits in problems is that it requires you to know or have a candidate for the limit L before you can formally verify a sequence converges One goal in calculus is to learn ways to calculate limits other than by means of this formal de nition Basic Rules about Limits Suppose x and y are both convergent sequences Then the following sequences are also convergent Zn 2 ax for any constant a x Zn 2 provided lim yn i 0 yn ngtoo Basic Rules about Limits Suppose 6n and yn are both convergent sequences In fact if lim 6n L and lim yn M then ngt00 ngt00 lim a xn a L for any constant a nHJroo 1imltxn yngtLM nHJroo lim xnyn 2 LM nHJroo lim x i provided M x O M nHJroo y n Example woo 107 lim 107 ngt00 rule for quotients Example hm 35X01n quotEgg 01 woo 107 lim 107 ngt00 lim 3 lim lt 5gtltO1 gt ngt00 ngt00 lim 1 lim 07 naoo naoo rule for sums Example woo 107 lim 107 ngt00 lim 3 lim lt 5gtltOl gt ngt00 ngt00 lim 1 lim 07 ngt00 ngt00 lim 3 5 lim 01 ngt00 ngt00 lim 1 lim 07 ngt00 ngt00 rule for constant multiples Example hm 3 5gtltO1 quot13303 5X01quot woo 107 lim 107 ngt00 lim 3 lim lt 5gtltO1 gt ngt00 ngt00 lim 1 lim 07 ngt00 ngt00 lim 3 5 lim 01 naoo lim 07 naoo we know these limits Example lim lt3 5gtltO1 naoo lim 107 naoo lim 3 lim lt 5gtltO1 gt naoo naoo lim naoo 3 5gtltO1 1O7 lim 1 lim 07quot naoo naoo 3 511m 01 naoo 1 lim 07 naoo Example hm 3 5gtltO1 quot13303 5X01quot woo 107 lim 107 ngt00 lim 3 lim lt 5gtltO1 gt ngt00 ngt00 lim 1 lim 07 ngt00 ngt00 1 HJFOO we know these limits Example lim lt3 5gtltO1 naoo lim 107 naoo lim 3 lim lt 5gtltO1 gt naoo naoo lim naoo 3 5gtltO1 1O7 lim 1 lim 07quot naoo naoo 3 511m 01 naoo 1 lim 07 ngt00 3 5gtltO 1 l O Example lim lt3 5gtltO1 naoo lim 107 naoo lim 3 lim lt 5gtltO1 gt naoo naoo lim naoo 3 5gtltO1 1O7 lim 1 lim 07quot naoo naoo 3 511m 01 naoo 1 lim 07 ngt00 3 5gtltO 10 2 3 Another Useful Rule Suppose x converges to L and f x is continuous at L Then lim naoo Another Useful Rule Suppose x converges to L and f x is continuous at L Then lim naoo Example lim coslt075 gt cosO 1 naoo Example densitv regulated growth We39ve seen that a population growing according the the law xnl r xn goes extinct converges to 0 if I lt l grows without bound diverges to 00 if I gt 1 If we want a model that describes a population that survives but does not grow without bound then r cannot remain constant over time Example densitv regulated growth We39ve seen that a population growing according the the law xnl r xn goes extinct converges to 0 if I lt l grows without bound diverges to 00 if I gt 1 If we want a model that describes a population that survives but does not grow without bound then r cannot remain constant over time A famous model replaces I by l lcxn 739 O lt C units l population number The growth factor decreases to 0 as the population size xquot increases Example densitV regulated growth The BevertonHolt Equation 1 x7 l l cxn xnl r Question What do solutions do when r gt 1 30 25 20 15 10 Example densitV regulated growth The BevertonHolt Equation 1 1 cxn xnl r xn Question What do solutions do when r gt 1 BevertonHolt r 12 c 001 With initial condition x0 l xquot appears to converge to a limit L m 20 10 15 20 25 30 35 40 45 50 Example densitV regulated growth The BevertonHolt Equation 1 l cxn xnl r xn Question What do solutions do when r gt 1 BevertonHolt r 12 c 001 With initial condition x0 5 xquot appears to converge to the same limit L m 20 30 25 20 Example densitV regulated growth The BevertonHolt Equation x n1 r xn l l cxn Question What do solutions do when r gt 1 BevertonHolt r 12 c 001 With initial condition x0 10 XM appears to converge to the same limit L m 20 30 o 25 20 Exarnple densitV regulated growth The BevertonHolt Equation x n1 r xn l l cxn Question What do solutions do when r gt 1 BevertonHolt r 12 c 001 With initial condition x0 30 XM appears to converge to the same limit L m 20 Example densitV regulated growth The BevertonHolt Equation 1 x x r n l l cxn n1 Question What do solutions do when r gt 1 BevertonHolt r 12 c 001 30 o 25 We conjecture that 20 for all initial conditions x0 gt 0 the solution sequence xn will to converge to the same limit L m 20 Example densitV regulated growth The BevertonHolt Equation 1 l cxn xnl r xn Question What do solutions do when r gt 1 We conjecture that for all initial conditions x0 gt 0 the solution sequence xn will converge to a limit L that depends on r and c Example densitV regulated growth The BevertonHolt Equation 1 l cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula xquot gn Example densitV regulated growth The BevertonHolt Equation 1 l cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula 7 1x0 r l 0x0 r quot 0x0 71 Example densitV regulated growth The BevertonHolt Equation 1 l cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula 7 1x0 quot r l 0x0 r quot 0x0 lim xquot lim 039 onn VHltgtO quotn00r 1 cx0r cx0 Use basic rules for taking limits Example densitV regulated growth The BevertonHolt Equation 1 l cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula 7 1x0 quot r l cx0r quot cx0 r l x 11m xn lim ngt 00 ngtoo r 6x0 0x0 Example densitV regulated growth The BevertonHolt Equation 1 l cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula 7 1x0 quot r l cx0r quot cx0 r l x 11m xn lim ngt 00 ngtoo r 6x0 0x0 NOTE lim r quot 71 00 Example densitV regulated growth The BevertonHolt Equation 1 1 cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula 7 1x0 quot r l 0x0 r quot 0x0 lim xquot lim 039 Dx n quotn00 quotn00r l cx0r cx0 71 NOTE lim 7 quot lim ngt 00 ngtoo r Example densitV regulated growth The BevertonHolt Equation 1 1 cxn n1 r xn Question What do solutions do when r gt 1 ngt 00 ngtoo 71 NOTE lim 7 quot lim Example densitV regulated growth The BevertonHolt Equation 1 1 cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula 7 1x0 quot r l 0x0 r quot 0x0 lim xquot lim 039 Dx n quotn00 quotn00r l cx0r cx0 71 NOTE lim 7 quot lim 0 ngt 00 ngtoo r Example densitV regulated growth The BevertonHolt Equation 1 1 cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula 7 1x0 quot r l 0x0 r quot 0x0 lim xquot lim 039 onn quotn00 quotn00r 1 cx0r cx0 7 1x0 0ch Example densitV regulated growth The BevertonHolt Equation 1 1 cxn xnl r xn Question What do solutions do when r gt 1 It turns out there is a solution formula r lx0 quot r l cx0r quot cx0 lim x lim PM quotn00 quot quotn00r l cx0r quotch 7 1x0 0ch r l C Example densitV regulated growth The BevertonHolt Equation 1 x x r n l l cxn n1 Question What do solutions do when r gt 1 r l lim xn 71 00 C BevertonHolt r 12 c 001 lim xn 20 quotn00 001 Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 a Number of bacteria in millions d antibiotic dose at hour 11 l in milligram a initial infection millions of S aureus Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 2a Recall Our Conjecture There is a threshold for the dosage 61 below Which the treatment does not cure the infection is and above Which it does The threshold depends on the initial infection number a Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 2a Mathematical Formulation of the Conjecture There is a threshold value of d below which the solution xn tends to 00 and above which X tends to oo Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x 06d x0 a To look for a solution formula the most basic rst step is write out enough terms at the beginning of the solution sequence until you see a pattern developing Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4xn O6d x0 2a x1 4x0 O6d 4a O6d Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4xn O6d x0 61 x1 4x0 O6d 4a O6d x2 4x1 O6d 42a 4x06d O6d Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4xn O6d x0 61 x1 4x0 O6d 4a O6d x2 4x1 O6d 42a 4x06d O6d x3 4x2 O6d 4301 42 x06d 4gtltO6d O6d Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 61 x1 4x0 O6d 4a O6d x2 4x1 O6d 42a 4x06d O6d x3 4x2 O6d 4301 42 x06d 4gtltO6d O6d x4 4x3 O6d 4401 43 gtltO6a 42 x06d 4gtltO6d O6d See a pattern Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 61 x1 4x0 O6d 4a O6d x2 4x1 O6d 42a 4x06d O6d x3 4x2 O6d 4301 42 x06d 4gtltO6d O6d x4 4x3 O6d 4401 43 gtltO6a 42 x06d 4gtltO6d O6d See a pattern A power of 4 building up Finite Geometric Series a short digression Finite Geometric Series a short digression 2 1 Sn a l ar l ar ar A sum of n terms Finite Geometric Series a short digression 2 1 Sn a l ar l ar ar A sum of n terms Each term is r times its predecessor Finite Geometric Series a short digression 2 1 Sn a l ar l ar ar A sum of n terms Each term is r times its predecessor Or equivalently the ratio of a term to its predecessor is the same number r for all terms Finite Geometric Series a short digression 2 1 Sn a l ar l ar ar A sum of n terms Each term is r times its predecessor Initial term a Finite Geometric Series a short digression Sn a l ar l ar2 ar There is a simpler formula available for calculating this sum Sn a l ar l ar2 ar l l Finite Geometric Series a short digression Sn a l ar l ar2 ar There is a simpler formula available for calculating this sum Sn a l ar l ar2 ar l l 2 1 rs ar l ar ar ar Subtract Sn rsnza ar l r Solvefor S 61 l r Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4xn O6d x0 2a x 4quota 14424quot106d n Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x 06d x0 a xquot 4quota lt1442 4quot 1O6d A nite geometric series of n terms With ratio r 4 and rst term a 1 Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 a xquot 4quota lt1442 4quot 1O6d A nite geometric series of n terms With ratio r 4 and rst term a 1 l rquot 1 4quot l r 1 4 S261 Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4xn O6d x0 2a x 4quota 14424quot106d n Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4xn O6d x0 2a x 4quota l4quot 106d 3 n Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 2a xquot 4quota lt4quot l 02d a solution formula Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 2a xquot 4quota 4quot 102d V number number per milligrams of bacteria milligram General Simulated Annealing Pseudocode Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x 06d x0 a xquot 4quota 4quot 102d Recall we are interested in What happens to xquot as n gt l 00 So let39s package the two 4 terms together Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x 06d x0 a xquot a 02d4quot 02d Recall we are interested in What happens to xquot as n gt l 00 So let39s package the two 4 terms together Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 61 xquot a 02d4quot 02d a 02dgt0 implies xn gtoo as n gtoo Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 61 xquot a 02d4quot 02d a 02dgt0 implies xn gtoo as n gtoo a O2dlt0 implies xn gt oo as n gtoo Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 2a xquot a 02d4quot 02d a dltE implies xn gtoo as n gtoo a dgtE implies xn gt oo as n gtoo Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 2a xquot a 02d4quot 02d a dltE implies xn gtoo as n gtoo dgt implies xn gt oo as n gtoo threshold for d numb er numbermg Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x O6d x0 2a xquot a 02d4quot 02d dlt5a implies xn gtoo as n gtoo dgt5a implies xn gt oo as n gtoo This shows the conjecture is true and that the threshold dosage is 5a mg Example Return to the model for an antibiotic treatment of an infection by Staphylococcus aureus an 4x 06d x0 a The threshold dosage is 5a mg This corroborates the two examples that led to the conjecture alanddl alandd5l threshold 5 threshold 5 treatment fails treatment succeeds In this example we encountered nite geometric series 2 1 Sn a l ar l ar ar These series arise in many applications Here s another Example Return to the antibiotic treatment problem In the previous example we assumed no carry over of antibiotic om one treatment period to the next Example Return to the antibiotic treatment problem In the previous example we assumed no carry over of antibiotic om one treatment period to the next Suppose instead that the antibiotic can stay in effect for longer than the time between treatments so there is an accumulation of antibiotic in the patient over time Example Return to the antibiotic treatment problem In the previous example we assumed no carry over of antibiotic om one treatment period to the next Suppose instead that the antibiotic can stay in effect for longer than the time between treatments so there is an accumulation of antibiotic in the patient over time We want to determine how much antibiotic is present at any time in order to make sure that in the long run it lies between acceptable limits Example Assume 95 of the antibiotic is used or otherwise eliminated between treatments Example Assume 95 of the antibiotic is used or otherwise eliminated between treatments 12 yn amount mg antibodic in patient at time n 0 3 yn 1 2 residual left from treatment at time n new dose Example Assume 95 of the antibiotic is used or otherwise eliminated between treatments 12 yn amount mg antibodic in patient at time n 0 3 yn 1 residual left from treatment at time n new dose yn1 00532 d Example Assume 95 of the antibiotic is used or otherwise eliminated between treatments 12 7 7 7 yn amount mg antibodic in patient at time n 0 3 yn 1 residual left from treatment at time n new dose yn1 00532 d Question What dosages 039 will in the long run keep yn between 250 300 mg Example yn1 00532 d yo 2 initial amount in patient we assume 0 Example yn1 00532 d yo 2 initial amount in patient we assume 0 y1 005y0 d Example yn1 00532 d yo 2 initial amount in patient we assume 0 y1 OOSyOdd Example yn1 00532 d yo 2 initial amount in patient we assume 0 y1 2005y0 dd y2 005y1 d Example yn1 00532 d yo 2 initial amount in patient we assume 0 y1 2005y0 dd y2 005y1d 005d d Example yn1 00532 d yo 2 initial amount in patient we assume 0 y1 2005y0 dd y2 005y1d 005d d y3 005322 d Example yn1 00532 d yo 2 initial amount in patient we assume 0 y1 2005y0 dd y2 005y1d 005d d y3 005322 d 0052d 005d d Example yn1 005 y d yo initial amount in patient we assume 0 y1 2005y0dd y2 005y1d 005dd y3 005322 d 0052d 005d d y4 005323 d Example yn1 00532 d yo 2 initial amount in patient we assume 0 y1 2005y0 dd y2 005y1d 005d d y3 005322 d 0052d 005d d y4 005323 d 0053d0052d005dd Example yn1 005yn d yo 2 initial amount in patient we assume 0 y1 2005y0 dd y2 005y1d 005d d y3 005322 d 0052d 005d d y4 005323 d 0053d0052d005dd observe pattern Example yn1 00532 d yo 2 initial amount in patient we assume 0 y1 2005y0 dd y2 005y1d 005d d y3 005322 d 0052d 005d d y4 005323 d 0053d0052d005dd yn 005 Hd 005 2d o 005d d mm yn d 005d 005 005 the solution formula is a finite geometric series mm yn d 005d 005 005 What happens in the long run lim y lim d 005d 005201 005 1d Vl gtoo ngt00 Example yn1 00532 d yn d 005d 005 005 What happens in the long run lim y lim d 005d 005201 005 1d 17 00 n gt oo lim d 005d 005201 005 1d 17 00 This is called an in nite series Example yn1 00532 d yn d 005d 005 005 What happens in the long run lim y lim d 005d 005201 005 1d n gt oo n gt oo lim d 005d 005201 005 1d 17 00 This is called an in nite series sn d 005d 0052d o 005 Hd The sequence Ofpariial sums Definition An in nite series lim a0 a1 a2 am converges if and only if 17 00 its sequence of partial sums Sn 2 a0 a1 a2 an1 converges Definition An in nite series lim a0 a1 a2 am converges if and only if 17 00 its sequence of partial sums Sn 2 a0 a1 a2 an1 converges Some notation Cloalaz an1a0ala2an 17 00 Definition An in nite series lim a0 a1 a2 am converges if and only if 17 00 its sequence of partial sums Sn 2 a0 a1 a2 an1 converges Some notation 17 00 lim a0a1a2an1a0ala2anZan n0 sigma notation Definition An in nite series 2 at converges if and only if i0 n l its sequence of partial sums Sn 2611 converges i0 Definition 00 An in nite series 2 at converges if and only if i0 n l its sequence of partial sums Sn 2611 converges i0 An in nite series 2 at diverges if and only if i0 n l its sequence of partial sums Sn 2611 diverges i0 mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run lim yn limd005d0052dm005quot1d Vl gtoo ngt00 mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run 17 00 Does this infinite series converge hm yn 2005 i0 If so to what mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run 17 00 hm yn 2005 i0 n gtoo 00 n l By definition 2005 lim 2005101 i0 z0 Example yn1 00532 d yn d 005d 0052d005n 1d What happens in the long run 17 00 lim yn Zoosid i0 00 n l By definition 2005 lim 2005101 i0 z0 n gtoo A finite geometric series Example yn1 00532 d yn d 005d 0052d005n 1d What happens in the long run 17 00 lim yn Zoosid i0 n gtoo Recall n A finite geometric series 00 n l By definition 2005 lim 2005101 10 10 mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run 17 00 hm yn 2005 i0 n gtoo 00 n l By definition 2005 lim 2005101 i0 z0 n gtoo mm yn1 005yn d ynd005d005 wuoode What happens in the long run 17 00 hm yn 2005 i0 00 n l By definition 2005 lim 2005101 i0 n gtoo i0 hm 011 005 new 1 005 1 1 005 d mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run 17 00 hm yn 2005 i0 n gtoo 00 n l By definition 2005 lim 2005101 i0 z0 n gtoo 1 005 95 20 d mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run hm yn 2005 2d i0 20 17 00 mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run hm yn 2005 2d i0 20 17 00 Question What dosages 039 will in the long run keep yn between 250 300 mg mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run 17 00 hm yn 2005 d i0 Question What dosages 039 will in the long run keep yn between 250 300 mg Answer 250 g 2d g 300 20 mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run 17 00 hm yn 2005 d i0 Question What dosages 039 will in the long run keep yn between 250 300 mg Answer 5000 6000 S d S 19 19 mm yn1 005yn d yn d 005d 0052d005n 1d What happens in the long run 17 00 hm yn 2005 d i0 Question What dosages 039 will in the long run keep yn between 250 300 mg Answer 5000 6000 S d S 19 19 2632mgz massing Not all infinite series of interest are geometric series There are two basic questions concerning an infinite series Not all infinite series of interest are geometric series There are two basic questions concerning an infinite series Does it converge Not all infinite series of interest are geometric series There are two basic questions concerning an infinite series Does it converge If so to what Not all infinite series of interest are geometric series There are two basic questions concerning an infinite series Does it converge If so to what For geometric series we know the answer to both iari converges to if lrl lt 1 iari diverges if lrl 21 i0 Not all infinite series of interest are geometric series There are two basic questions concerning an infinite series Does it converge If so to what First question is studied in Chapter 9 Not all infinite series of interest are geometric series There are two basic questions concerning an infinite series Does it converge If so to what First question is studied in Chapter 9 Second question is in general much more difficult Sorne relevant methods of calculation are studied in Chapter 10

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