11.1 Intro to Sequences Series (1)

7/23/2019 11.1 Intro to Sequences Series (1)

1/21

11.1 An Introduction to11.1 An Introduction toSequences & SeriesSequences & Series

p. 651


2/21

What is a sequence?

What is the difference betweenfinite and infinite?


3/21

SequenceSe

quence:: A list of ordered numbers separated byA list of ordered numbers separated by

commas.commas.

Each number in the list is called aEach number in the list is called a termterm..

For Eample:For Eample:

Sequence 1Se

quence 1 Sequence !Se

quence !

!"#"6"$"1%!"#"6"$"1% !"#"6"$"1%"!"#"6"$"1%"

'erm 1" !" (" #" 5'erm 1" !" (" #" 5'erm 1" !" (" #" 5'erm 1" !" (" #" 5

)omain)omain* relati+e position of each term ,1"!"("#"5-* relati+e position of each term ,1"!"("#"5-sually be/ins 0ith position 1 unless other0isesually be/ins 0ith position 1 unless other0isestated.stated.

an/ean

/e* the actual terms of the sequence* the actual terms of the sequence,!"#"6"$"1%-,!"#"6"$"1%-


4/21

Sequence 1Se

quence 1 Sequence !S

equence !

!"#"6"$"1%!"#"6"$"1% !"#"6"$"1%"!"#"6"$"1%"

A sequence can beA sequence can be finitefiniteoror infiniteinfinite..

'he sequence has'he sequence hasa last term ora last term or finalfinal

term.term.

,such as seq. 1-,such as seq. 1-

'he sequence'he sequencecontinues 0ithoutcontinues 0ithout

stoppin/.stoppin/.

,such as seq. !-,such as seq. !-

2oth sequences ha+e a2oth sequences ha+e a /eneral rule

/eneral rule: a: ann3 !n 0here3 !n 0here

n is the term 4 and an is the term 4 and annis the nth term.is the nth term.

'he /eneral rule can also be 0ritten in'he /eneral rule can also be 0ritten in functionfunction

notationnotation: f,n- 3 !n: f,n- 3 !n


5/21

Eamples:Eam

ples:

rite the first 6rite the first 6

terms of aterms of ann35n.35n. aa113513#3513#

aa!!35!3(35!3(

aa((35(3!35(3! aa##35#3135#31

aa553553%3553%

aa663563135631

#"("!"1"%"1#"("!"1"%"1

rite the first 6rite the first 6terms of aterms of ann3!3!nn..

aa113!3!113!3!

aa!!

3!3!!!

3#3#

aa((3!3!((3$3$

aa##3!3!##316316

aa55

3!3!55

3(!3(!

aa663!3!6636#36#

!"#"$"16"(!"6#!"#"$"16"(!"6#


6/21

EamplesEam

ples: rite a rule for the nth term.: rite a rule for the nth term.

'he seq. can be'he seq. can be0ritten as:0ritten as:

7r" a7r" ann3!8,53!8,5nn--

'he seq. can be'he seq. can be

0ritten as:0ritten as:

!,1-91" !,!-91" !,(-91"!,1-91" !,!-91" !,(-91"

!,#-91"!,#-91"

7r" a7r" ann3!n913!n91

,...625

2,125

2,25

2,5

2.a

,...5

2,

5

2,

5

2,

5

24321

,...9,7,5,3.b


7/21

Eample: 0rite a rule for the nth term.

!"6"1!"!%"!"6"1!"!%"

an be 0ritten as:an be 0ritten as:

1,!-" !,(-" (,#-" #,5-"1,!-" !,(-" (,#-" #,5-"

7r" a7r" ann3n,n91-3n,n91-


8/21

;raphin/ a Sequence;raphin/ a Sequence

'hin< of a sequence as ordered pairs for'hin< of a sequence as ordered pairs for/raphin/. ,n " a/raphin/. ,n " ann--

For eample: ("6"="1!"15For eample: ("6"="1!"15

0ould be the ordered pairs ,1"(-" ,!"6-"0ould be the ordered pairs ,1"(-" ,!"6-",("=-" ,#"1!-" ,5"15- /raphed li


9/21

Graphing

na

1

(

!

6

(

=

#

1!


10/21

What is a sequence?

A collections of objects that is ordered so thatthere is a 1st, 2nd, 3rd, eber!

What is the differencebetween finite andinfinite?

"inite eans there is a last ter! #nfiniteeans the sequence continues withoutstopping!


11/21

Assi/nment:Assi/nment:

p. 655

=!= all"

s


12/21

$equences and $eries %a& 2 hat is a series?

@o0 do you


13/21

SeriesSeries

'he sum of the terms in a sequence.'he sum of the terms in a sequence.

an be finite or infinitean be finite or infinite

For Eample:For Eample:

Finite Seq.Finite Seq.

Infinite Seq.Infinite Seq.

!"#"6"$"1%!"#"6"$"1% !"#"6"$"1%"!"#"6"$"1%"

Finite SeriesFinite Series Infinite SeriesInfinite Series!9#969$91%!9#969$91% !9#969$91%9!9#969$91%9


14/21

Summation otationSummation otation Also calledAlso called sigma notationsigma notation

,si/ma is a ;ree< letter,si/ma is a ;ree< letter BBmeanin/ CsumD-meanin/ CsumD-

'he series !9#969$91% can be 0ritten as:'he series !9#969$91% can be 0ritten as:

i is called thei is called the index of summationindex of summation

,its ust li


15/21

Summation otation for anSummation otation for an

Infinite SeriesInfinite Series

Summation notation for the infinite series:Summation notation for the infinite series:

!9#969$91%9 0ould be 0ritten as:!9#969$91%9 0ould be 0ritten as:

2ecause the series is infinite" you must use i2ecause the series is infinite" you must use ifrom 1 to infinity ,from 1 to infinity ,G- instead of stoppin/ atG- instead of stoppin/ at

the 5the 5ththterm li


16/21

Eamples: rite each series inEamples: rite each series in

summation notation.summation notation.

a. #9$91!991%%a. #9$91!991%%

otice the series canotice the series can

be 0ritten as:be 0ritten as:

#,1-9#,!-9#,(-99#,!5-#,1-9#,!-9#,(-99#,!5-

7r #,i- 0here i /oes7r #,i- 0here i /oes

from 1 to !5.from 1 to !5.

otice the seriesotice the series

can be 0ritten as:can be 0ritten as:

25

1

4i

...5

4

4

3

3

2

2

1

. ++++b

...14

413

312

211

1 ++

++

++

++

.to1fromgoeswhere

1

Or, +

ii

i

+1 1i

i


17/21

EampleEample: Find the sum of the: Find the sum of the

series.series.

< /oes from 5 to 1%.< /oes from 5 to 1%.

,5,5!!91-9,691-9,6!!91-9,H91-9,H!!91-9,$91-9,$!!91-9,=91-9,=!!91-9,1%91-9,1%!!91-91-

3 !69(H95%9659$!91%13 !69(H95%9659$!91%1

33

(61(61

+10

5

2

1k


18/21

Special Formulas ,shortcuts-Special Formulas ,shortcuts-

n

n

i

==1

12

)1(

1

+=

=

nni

n

i

6

)12)(1(

1

2 ++

==

nnni

n

i

1

n

i

c cn

=

=


19/21

Eample: Find the sum.Eample: Find the sum.

se the (se the (rdrd

shortcutshortcut

=

10

1

2

i

i

6

)12)(1( ++ nnn

6

)110*2)(110(10 ++=

6

21*11*10= 385

6

2310==


20/21

What is a series?

A series occurs when the ters of a sequence areadded!

'ow do &ou (now the difference between asequence and a series?

)he plus signs

What is siga notation?*

'ow do &ou write a series with suationnotation?

+se the siga notation with the pattern rule!

ae 3 forulas for special series!

1

n

i

c cn

=

=2

)1(

1

+=

=

nni

n

i 6

)12)(1(

1

2 ++=

=

nnni

n

i


21/21

Assi/nment:Assi/nment:

p! -..3/0-1 eer& 3rd

proble

11.1 Intro to Sequences Series (1)

Documents

Transcript of 11.1 Intro to Sequences Series (1)