Section 6.4: Series. Section 6.4 Series 413

Transcription

1 ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest $50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they have saved whe their daughter is ready to start college i 6 years? I this sectio, we will lear how to aswer this questio To do so, we eed to cosider the amout of moey ivested ad the amout of iterest eared Usig ummatio Notatio To fid the total amout of moey i the college fud ad the sum of the amouts deposited, we eed to add the amouts deposited each moth ad the amouts eared mothly The sum of the terms of a sequece is called a series Cosider, for example, the followig series The thpartial sum of a series is the sum of a fiite umber of cosecutive terms begiig with the first term The otatio represets the partial sum ummatio otatio is used to represet series ummatio otatio is ofte ow as sigma otatio because it uses the Gree capital letter sigma,, to represet the sum ummatio otatio icludes a explicit formula ad specifies the first ad last terms i the series A explicit formula for each term of the series is give to the right of the sigma A variable called the idex of summatio is writte below the sigma The idex of summatio is set equal to the lower limit of summatio, which is the umber used to geerate the first term i the series The umber above the sigma, called the upper limit of summatio, is the umber used to geerate the last term i a series If we iterpret the give otatio, we see that it ass us to fid the sum of the terms i the series a for through = 5 We ca begi by substitutig the terms for ad listig out the terms of this series a () a a a a 4 5 () 4 () 6 (4) 8 (5) 0

2 44 Chapter 6 We ca fid the sum of the series by addig the terms: ummatio Notatio The sum of the first terms of a series ca be expressed i summatio otatio as follows: a This otatio tells us to fid the sum of a from to = is called the idex of summatio, is the lower limit of summatio, ad is the upper limit of summatio The lower limit of summatio does ot have to be The lower limit of summatio ca be ay umber, but is frequetly used We will loo at examples with lower limits of summatio other tha Give summatio otatio for a series, evaluate the value Idetify the lower limit of summatio Idetify the upper limit of summatio ubstitute each value of from the lower limit to the upper limit ito the formula 4 Add to fid the sum Example (video example here) Evaluate 7 Accordig to the otatio, the lower limit of summatio is ad the upper limit is 7 o we eed to fid the sum of from to = 7We fid the terms of the series by substitutig =, 4, 5, 6, ad 7 ito the fuctio We add the terms to fid the sum

3 ectio 64 eries 45 Try it Now Evaluate 5 ( ) Writig Formulas for equeces ad Expressig eries usig ummatio Notatio that are ot ecessarily Arithmetic or Geometric The followig video shows two examples o how to express a series usig summatio otatio Oe of the series is either Arithmetic or Geometric Video Example: Two Examples of Expressig a eries usig ummatio Notatio The followig video shows how to fid a explicit formula for a sequece that is either arithmetic or geometric Video Example : Fidig a Explicit Formula for a equece The followig video shows two examples o how to express a series usig summatio otatio The series are either Arithmetic or Geometric Video Example : Expressig a eries usig ummatio Notatio Usig the Formula for Arithmetic eries Just as we studied special types of sequeces, we will loo at special types of series Recall that a arithmetic sequece is a sequece i which the differece betwee ay two cosecutive terms is the commo differece, d The sum of the terms of a arithmetic sequece is called a arithmetic series We ca write the sum of the first terms of a arithmetic series as: = a + (a + d) + (a + d) + + (a d) + a We ca also reverse the order of the terms ad write the sum as = a + (a d) + (a d) + + (a + d) + a If we add these two expressios for the sum of the first terms of a arithmetic series, we ca derive a formula for the sum of the first terms of ay arithmetic series a ( a d) ( a d) ( a d) a a ( a d) ( a d) ( a d) a ( a a ) ( a a ) ( a a )

4 46 Chapter 6 Because there are terms i the series, we ca simplify this sum to = (a + a ) We divide by to fid the formula for the sum of the first terms of a arithmetic series ( a a) Formula for the um of the First Terms of a Arithmetic eries A arithmetic series is the sum of the terms of a arithmetic sequece The formula for the sum of the first terms of a arithmetic sequece is: ( a a) Give terms of a arithmetic series, fid the sum of the first terms Idetify a ad a Determie a a ubstitute values for a, a, ad ito the formula 4 implify to fid Example Fid the sum of each arithmetic series a b c 8 a We are give a 5 ad a = Cout the umber of terms i the sequece to fid =0 ubstitute values for a, a, ad ito the formula ad simplify a a b We are give a 0 ad a = 50

5 ectio 64 eries 47 Use the formula for the geeral term of a arithmetic sequece to fid a a ( ) d 50 0 ( )( 5) 70 ( )( 5) 4 5 ubstitute values for a,, a ad ito the formula ad simplify 5 a a c To fid a, substitute ito the give explicit formula a 8 a () 8 5 We are give that =To fid a,substitute ito the give explicit formula 8 a a () 8 8 ubstitute values for a,, a a a ad ito the formula ad simplify Try it Now Use the formula to fid the sum of each arithmetic series a b c 0 5 6

6 48 Chapter 6 Example O the uday after a mior surgery, a woma is able to wal a half-mile Each uday, she wals a additioal quarter-mile After 8 wees, what will be the total umber of miles she has waled? This problem ca be modeled by a arithmetic series with a ad d = We are 4 looig for the total umber of miles waled after 8 wees, so we ow that = 8, ad we are looig for 8 To fid a 8, we ca use the explicit formula for a arithmetic sequece a a d( ) a 8 9 (8 ) 4 4 We ca ow use the formula for arithmetic series a a he will have waled a total of miles Try it Now A ma ears $00 i the first wee of Jue Each wee, he ears $50 more tha the previous wee After wees, how much has he eared? Usig the Formula for Geometric eries Just as the sum of the terms of a arithmetic sequece is called a arithmetic series, the sum of the terms i a geometric sequece is called a geometric series Recall that a geometric sequece is a sequece i which the ratio of ay two cosecutive terms is the commo ratio, r We ca write the sum of the first terms of a geometric series as = a + ra + r a + + r a Just as with arithmetic series, we ca do some algebraic maipulatio to derive a formula for the sum of the first terms of a geometric series We will begi by multiplyig both sides of the equatio by r r ra r a r a r a

7 ectio 64 eries 49 Next, we subtract this equatio from the origial equatio a ra r a r a r ra r a r a r a ( r) a r a Notice that whe we subtract, all but the first term of the top equatio ad the last term of the bottom equatio cacel out To obtai a formula for, divide both sides by ( r) a r r, r Formula for the um of the First Terms of a Geometric eries A geometric series is sum of the terms i a geometric sequece The formula for the sum of the first terms of a geometric sequece is represeted as a r r, r Give a geometric series, fid the sum of the first terms Idetify a, r, ad ubstitute values for a, r, ad ito the formula = a ( r ) r implify to fid Example 4 Use the formula to fid the idicated partial sum of each geometric series for the series a b 6 a a = 8, ad we are give that = We ca fid r by dividig the secod term of the series by the first 4 r 8 ubstitute values for a, r, ad ito the formula ad simplify a r r

8 40 Chapter Fid a by substitutig ito the give explicit formula a 6 We ca see from the give explicit formula that r = The upper limit of summatio is 6, so = 6 ubstitute values for a, r, ad ito the formula, ad simplify 6 a r r Try it Now 4 Use the formula to fid the idicated partial sum of each geometric series for the series, a 0 b 8 Example 5 At a ew job, a employee s startig salary is $6,750 He receives a 6% aual raise Fid his total earigs at the ed of 5 years The problem ca be represeted by a geometric series with a 6,750; 5; ad r 06 ubstitute values for a, r, ad ito the formula ad simplify to fid the total amout eared at the ed of 5 years 5 r r 5 a 6, , 0990 He will have eared a total of $8,0990 by the ed of 5 years

9 ectio 64 eries 4 Try it Now 5 At a ew job, a employee s startig salary is $,00 he receives a % aual raise How much will she have eared by the ed of 8 years? Usig the Formula for the um of Ifiite Geometric eries Thus far, we have looed oly at fiite series ometimes, however, we are iterested i the sum of the terms of a ifiite sequece rather tha the sum of oly the first terms A ifiite series is the sum of the terms of a ifiite sequece A example of a ifiite series is 4 68 This series ca also be writte i summatio otatio as, where the upper limit of summatio is ifiity Because the terms are ot tedig to zero, the sum of the series icreases without boud as we add more terms Therefore, the sum of this ifiite series is ot defied Whe the sum is ot a real umber, we say the series diverges Determiig Whether the um of a Ifiite Geometric eries is Defied If the terms of a ifiite geometric series approach 0, the sum of a ifiite geometric series ca be defied The terms i this series approach 0: The commo ratio r = 0 As gets very large, the values of r get very small ad approach 0 Each successive term affects the sum less tha the precedig term As each succeedig term gets closer to 0, the sum of the terms approaches a fiite value The terms of ay ifiite geometric series with r approach 0; the sum of a geometric series is defied whe < r < Determiig Whether the um of a Ifiite Geometric eries is Defied A sum of a ifiite series is defied if the series is geometric ad < r < Give the first several terms of a ifiite series, determie if the sum of the series exists Fid the ratio of the secod term to the first term Fid the ratio of the third term to the secod term Cotiue this process to esure the ratio of a term to the precedig term is costat throughout If so, the series is geometric 4 If a commo ratio, r, was foud i step, chec to see if r If so, the sum is defied If ot, the sum is ot defied

10 4 Chapter 6 Example 6 Determie whether the sum of each ifiite series is defied a b 4 c d 7 5 a The ratio of the secod term to the first is, which is ot the same as the ratio of the third term to the secod, The series is ot geometric b The ratio of the secod term to the first is the same as the ratio of the third term to the secod The series is geometric with a commo ratio of The sum of the ifiite series is defied c The give formula is expoetial with a base of ; the series is geometric with a commo ratio of The sum of the ifiite series is defied d The give formula is ot expoetial; the series is ot geometric because the terms are icreasig, ad so caot yield a fiite sum Try it Now 6 Determie whether the sum of the ifiite series is defied a b c 5 ( 0)

11 ectio 64 eries 4 Fidig ums of Ifiite eries Whe the sum of a ifiite geometric series exists, we ca calculate the sum The formula for the sum of a ifiite series is related to the formula for the sum of the first terms of a geometric series a r r We will examie a ifiite series with r = What happes to r as icreases? The value of 0, 04 0, 048,576 0, 07, 74,84 r decreases rapidly What happes for greater values of? As gets very large, r gets very small We say that, as icreases without boud, r approaches 0 As r approaches 0, r approaches Whe this happes, the umerator approaches a This give us a formula for the sum of a ifiite geometric series: a r Formula for the um of a Ifiite Geometric eries The formula for the sum of a ifiite geometric series with < r < is a r Give a ifiite geometric series, fid its sum a Idetify a ad r b Cofirm that < r < c ubstitute values for a ad r ito the formula, = a r d implify to fid

12 44 Chapter 6 Example 7 Fid the sum, if it exists, for the followig: a b c d 4, a There is ot a costat ratio; the series is ot geometric b There is a costat ratio; the series is geometric a 486 ad r = 9944 = 04, so 486 the sum exists ubstitute a 486 ad r 04 ito the formula ad simplify to fid the sum: = a r c The formula is expoetial, so the series is geometric with r = Fid a by substitutig ito the give explicit formula: a 4,74 4,74 ubstitute a 4,74 ad r ito the formula, ad simplify to fid the sum: = a r 4,74, 805 d The formula is expoetial, so the series is geometric, but r > The sum does ot exist

13 ectio 64 eries 45 Example 8 Fid a equivalet fractio for the repeatig decimal 0 We otice the repeatig decimal 0 0 so we ca rewrite the repeatig decimal as a sum of terms Looig for a patter, we rewrite the sum, oticig that we see the first term multiplied to 0 i the secod term, ad the secod term multiplied to 0 i the third term 0 0 (0)( 0 ) (0)(0)(0) First term ecodterm Notice the patter; we multiply each cosecutive term by a commo ratio of 0 startig with the first term of 0 o, substitutig ito our formula for a ifiite geometric sum, we have a 0 0 r 0 09 Try it Now 7 Fid the sum, if it exists a 9 b c olvig Auity Problems At the begiig of the sectio, we looed at a problem i which a couple ivested a set amout of moey each moth ito a college fud for six years A auity is a ivestmet i which the purchaser maes a sequece of periodic, equal paymets To fid the amout of a auity, we eed to fid the sum of all the paymets ad the iterest eared I the example, the couple ivests $50 each moth This is the value of the iitial deposit The accout paid 6% aual iterest, compouded mothly To fid the iterest rate per paymet period, we eed to divide the 6% aual percetage iterest (APR) rate by o the mothly iterest rate is 05% We ca multiply the amout i the accout each moth by 005% to fid the value of the accout after iterest has bee added We ca fid the value of the auity right after the last deposit by usig a geometric series with a 50 ad r =005% =005 After the first deposit, the value of the auity will be $50 Let us see if we ca determie the amout i the college fud ad the iterest eared

14 46 Chapter 6 We ca fid the value of the auity after deposits usig the formula for the sum of the first terms of a geometric series I 6 years, there are 7 moths, so = 7 We ca substitute a = 50, r =005, ad = 7 ito the formula, ad simplify to fid the value of the auity after 6 years: , After the last deposit, the couple will have a total of $4,044 i the accout Notice, the couple made 7 paymets of $50 each for a total of 7(50) = $,600 This meas that because of the auity, the couple eared $7044 iterest i their college fud Give a iitial deposit ad a iterest rate, fid the value of a auity Determie a, the value of the iitial deposit Determie, the umber of deposits Determie r 4 Divide the aual iterest rate by the umber of times per year that iterest is compouded 5 Add to this amout to fid r 6 ubstitute values for a, r, ad ito the formula for the sum of the first terms of a geometric series, = a ( r ) r 7 implify to fid, the value of the auity after deposits Example 9 A deposit of $00 is placed ito a college fud at the begiig of every moth for 0 years The fud ears 9% aual iterest, compouded mothly, ad paid at the ed of the moth How much is i the accout right after the last deposit? The value of the iitial deposit is $00, so a =00 A total of 0 mothly deposits are made i the 0 years, so =0 To fid r, divide the aual iterest rate by to fid the mothly iterest rate ad add to represet the ew mothly deposit 009 r 0075 ubstitute a =00, r =0075, ad =0 ito the formula for the sum of the first terms of a geometric series, ad simplify to fid the value of the auity , o the accout has $9,54 after the last deposit is made

15 ectio 64 eries 47 Try it Now 8 At the begiig of each moth, $00 is deposited ito a retiremet fud The fud ears 6% aual iterest, compouded mothly, ad paid ito the accout at the ed of the moth How much is i the accout if deposits are made for 0 years? Try it Now Aswers 8 a 64 b 8 c 80 $,05 4 a,00000 b 9,840 5 $75,5 6a The sum is ot defied It is geometric with 6b The sum of the ifiite series is defied 6c The sum of the ifiite series is defied 7a b The series is ot geometric c 8 $9,4088 r

16 48 Chapter 6 ectio 64 Exercises th What is a partial sum? What is the differece betwee a arithmetic sequece ad a arithmetic series? What is a geometric series? 4 How is fidig the sum of a ifiite geometric series differet from fidig the th partial sum? 5 What is a auity? For the followig exercises, express each descriptio of a sum usig summatio otatio 6 the sum of terms m + m from m = to = 5 7 the sum from = 0 to = 4 of 5 8 the sum of 6 5 from = to = 9 the sum of addig the umber 4 five times For the followig exercises, express each arithmetic series usig summatio otatio For the followig exercises, use the formula for the sum of the first terms of each arithmetic series For the followig exercises, express each geometric series usig summatio otatio For the followig exercises, use the formula for the sum of the first terms of each geometric series, ad the state the idicated sum (0) a a 9 0 5()

17 ectio 64 eries 49 For the followig exercises, determie whether the ifiite series has a sum If so, write the formula for the sum If ot, state the reaso m m 5 For the followig exercises, use the iformatio give below Javier maes mothly deposits ito a savigs accout He opeed the accout with a iitial deposit of $50 Each moth thereafter he icreased the previous deposit amout by $0 6 Graph the arithmetic sequece showig oe year of Javier s deposits 7 Graph the arithmetic series showig the mothly sums of oe year of Javier s deposits For the followig exercises, use the geometric series 8 Graph the first 7 partial sums of the series 9 What umber does seem to be approachig i the graph from Exercise 9? Fid the sum to explai why this maes sese For the followig exercises, fid the idicated sum 4 0 a a 6 ( ) 7 7 For the followig exercises, use the formula for the sum of the first terms of a arithmetic series to fid the sum

18 40 Chapter 6 9 For the followig exercises, use the formula for the sum of the first terms of a geometric series to fid the partial sum 40 6 for the series for the series For the followig exercises, fid the sum of the ifiite geometric series (05) For the followig exercises, determie the value of the auity for the idicated mothly deposit amout, the umber of deposits, ad the iterest rate 48 deposit amout: $50; total deposits: 60; iterest rate: 5%, compouded mothly 49 deposit amout: $50; total deposits: 4; iterest rate: %, compouded mothly 50 deposit amout: $450; total deposits: 60; iterest rate: 45%, compouded quarterly 5 deposit amout: $00; total deposits: 0; iterest rate: 0%, compouded semiaually 5 Keisha devised a wee-log study pla to prepare for fials O the first day, she plas to study for hour, ad each successive day she will icrease her study time by 0 miutes How may hours will Keisha have studied after oe wee? 5 A boulder rolled dow a moutai, travelig 6 feet i the first secod Each successive secod, its distace icreased by 8 feet How far did the boulder travel after 0 secods? 54 A scietist places 50 cells i a petri dish Every hour, the populatio icreases by 5% What will the cell cout be after day?

19 ectio 64 eries 4 55 A pedulum travels a distace of feet o its first swig O each successive swig, it travels 4 the distace of the previous swig What is the total distace traveled by the pedulum whe it stops swigig? 56 Rachael deposits $,500 ito a retiremet fud each year The fud ears 8% aual iterest, compouded mothly If she opeed her accout whe she was 9 years old, how much will she have by the time she s 55? How much of that amout was iterest eared?