Intro to Series and Sequences

7/29/2019 Intro to Series and Sequences

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11.1 An Introduction toSequences & Series

p. 651


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Sequence:

A list of ordered numbers separated bycommas.

Each number in the list is called a term.

For Example:

Sequence 1 Sequence 2

2,4,6,8,10 2,4,6,8,10,

Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5

Domain relative position of each term (1,2,3,4,5)Usually begins with position 1 unless otherwisestated.

Range the actual terms of the sequence(2,4,6,8,10)


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Sequence 1 Sequence 2

2,4,6,8,10 2,4,6,8,10,

A sequence can be finiteor infinite.

The sequence hasa last term or finalterm.

(such as seq. 1)

The sequencecontinues withoutstopping.

(such as seq. 2)

Both sequences have a general rule: an = 2n wheren is the term # and an is the nth term.

The general rule can also be written in function

notation: f(n) = 2n


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Examples:

Write the first 6

terms of an=5-n. a1=5-1=4

a2=5-2=3

a3=5-3=2

a4=5-4=1

a5=5-5=0

a6=5-6=-1

4,3,2,1,0,-1

Write the first 6terms of an=2n.

a1=21=2

a2=22=4

a3=23=8

a4=24=16

a5=25=32

a6=26=64

2,4,8,16,32,64


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Examples: Write a rule for the nth term.

The seq. can bewritten as:

Or, an=2/(5n)

The seq. can be

written as:

2(1)+1, 2(2)+1, 2(3)+1,2(4)+1,

Or, an=2n+1

,...625

2,

125

2,

25

2,

5

2.a

,...5

2,

5

2,

5

2,

5

24321

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


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Example: write a rule for the nth term.

2,6,12,20,

Can be written as:

1(2), 2(3), 3(4), 4(5),

Or, an=n(n+1)


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Graphing a Sequence

Think of a sequence as ordered pairs forgraphing. (n , an)

For example: 3,6,9,12,15

would be the ordered pairs (1,3), (2,6),(3,9), (4,12), (5,15) graphed like points in a

scatter plot* Sometimes it helps to find the rule first

when you are not given every term in afinite sequence.

Term # Actual term


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Series

The sum of the terms in a sequence.

Can be finite or infinite

For Example:

Finite Seq. Infinite Seq.

2,4,6,8,10 2,4,6,8,10,

Finite Series Infinite Series2+4+6+8+10 2+4+6+8+10+


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Summation Notation Also called sigma notation

(sigma is a Greek letter meaning sum)

The series 2+4+6+8+10 can be written as:

i is called the index of summation

(its just like the n used earlier).Sometimes you will see an n or k here instead of i.

The notation is read:

the sum from i=1 to 5 of 2i

5

12i

i goes from 1

to 5.


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Summation Notation for anInfinite Series

Summation notation for the infinite series:

2+4+6+8+10+ would be written as:

Because the series is infinite, you must use ifrom 1 to infinity () instead of stopping at

the 5th term like before.

1

2i


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Examples: Write each series insummation notation.

a. 4+8+12++100

Notice the series canbe written as:

4(1)+4(2)+4(3)++4(25)

Or 4(i) where i goesfrom 1 to 25.

Notice the seriescan be written 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


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Example: Find the sum of theseries.

k goes from 5 to 10.

(52+1)+(62+1)+(72+1)+(82+1)+(92+1)+(102+1)

= 26+37+50+65+82+101

= 361

10

5

2

1k


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Special Formulas (shortcuts!)

n

n

i

1

12

)1(

1

nni

n

i

6

)12)(1(

1

2

nnni

n

i


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Example: Find the sum.

Use the 3rd

shortcut!

10

1

2

i

i

6

)12)(1( nnn

6

)110*2)(110(10

6

21*11*10 385

6

2310


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Assignment
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Intro to Series and Sequences

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