chapter 6: series and sequences...the infinite sequence an infinite sequence is a function whose...
TRANSCRIPT
![Page 1: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/1.jpg)
Chapter 6:
Series and SequencesSSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
![Page 2: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/2.jpg)
Chapter 6: Series and Sequences
Lecture 22: Series
Lecture 23: Sequences
![Page 3: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/3.jpg)
Lecture 21: Sequences
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
![Page 4: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/4.jpg)
Sequences
• A sequence is a function whose domain is the set
ℤ+ ={1, 2, 3, …} of positive integers and whose range is the
set ℝ of real number.
![Page 5: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/5.jpg)
Two Kinds of Number Sequence
1. Finite Sequence; and2. Infinite Sequence.
![Page 6: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/6.jpg)
The Infinite Sequence
• An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no
last term.
![Page 7: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/7.jpg)
The Finite Sequence
•A finite sequence is function whose domain consist of the first n positive integers only. In other words, it is a sequence having a
last term.
![Page 8: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/8.jpg)
Classroom Task 1:
Direction: Identify whether the
following are finite or infinite sequence. Clap three
times if it is FINITE and once if it is INFINITE.
![Page 9: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/9.jpg)
Example 1:
Finite or Infinite?
,...36,25,16,9,4,1
![Page 10: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/10.jpg)
Example 2:
Finite or Infinite?
7,6,5,4,3,2,1
![Page 11: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/11.jpg)
Example 3:
Finite or Infinite?
,...15,12,9,6,3
![Page 12: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/12.jpg)
Example 4:
Finite or Infinite?
14,12,10,8,6,4,2
![Page 13: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/13.jpg)
Example 5:
Finite or Infinite?
15,13,11,9,7,5,3,1
![Page 14: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/14.jpg)
Example 6:
Finite or Infinite?
,...13,8,5,3,2,1,1
![Page 15: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/15.jpg)
Example 7:
Write the first five terms of the sequence whose nth term is
given by the formula:
.23 nan
![Page 16: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/16.jpg)
Final Answer:The first term of the sequence is 5, the second term is 8, the third term
is 11, and so on. The sequence can be also denoted as:
.17,14,11,8,5
![Page 17: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/17.jpg)
Example 8:
List the four terms of the infinite sequence whose nth
term is given by:
.)3()1( 2 nc n
n
![Page 18: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/18.jpg)
Final Answer:
The first four terms of the sequence are
.1,0,1,4
![Page 19: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/19.jpg)
Example 9:
Find the 9th term and the 21st
term of the sequence whose nth
term is:
.12
2
n
nan
![Page 20: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/20.jpg)
Final Answer:
The 9th term is 81/80and the 21st term is
441/440.
![Page 21: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/21.jpg)
Performance Task 1:
Please download, print
and answer the “Let’s
Practice 21.” Kindly work
independently.
![Page 22: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/22.jpg)
Lecture 22: SeriesSSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
![Page 23: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/23.jpg)
SeriesThe indicated sum of the terms of a
finite sequence a1, a2,a3, …, an is called series or finite series and
is denoted by:
....321 nn aaaaS
![Page 24: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/24.jpg)
Example 10:
Find S5 where:
.423 nan
![Page 25: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/25.jpg)
Final Answer:
S5 is 55.
![Page 26: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/26.jpg)
Example 11:
Find S4 where :
.)1( 2na n
n
![Page 27: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/27.jpg)
Final Answer:
S4 is 10.
![Page 28: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/28.jpg)
Did you know?
To describe the sum of the terms of a sequence, we use the
summation notation. The Greek letter Σ (sigma) is used to indicate
each sum.
![Page 29: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/29.jpg)
Two Kinds of Series
1. Finite Series; and2.Infinite Series
![Page 30: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/30.jpg)
Finite Series
The sum of the first n terms of the sequence is called a finite series or the nth partial sum of
the sequence.
![Page 31: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/31.jpg)
Finite Series
It is denoted by:
nn
n
i
i aaaaaaa
14321
1
...
![Page 32: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/32.jpg)
Finite Series
The sum of the first n terms of the sequence is called a finite series or the nth partial sum of
the sequence.
![Page 33: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/33.jpg)
Take Note:
i is called the index of the summation, n is the upper
limit of the summation, and 1is the lower limit of the
summation.
![Page 34: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/34.jpg)
Something to think about…
What is the correct way of reading the summation
notation? Let’s say
n
i
ia1
![Page 35: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/35.jpg)
Did you know?
is read as “the summation of ai where i
is from 1 to n”.
n
i
ia1
![Page 36: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/36.jpg)
Infinite Series
The sum of all terms of the infinite sequence is called a infinite series.
![Page 37: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/37.jpg)
Infinite Series
It is denoted by:
nn
i
i aaaaaaa
14321
1
...
![Page 38: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/38.jpg)
Take Note:
The upper limit summation of the finite series is n,
while in infinite series is ∞(symbol for infinity).
![Page 39: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/39.jpg)
Example 12:
Write out the terms of the following series:
5
1
22i
i
![Page 40: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/40.jpg)
Final Answer:
11025
1
2 i
i
![Page 41: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/41.jpg)
Example 13:
Write out the terms of the following series:
n
i
i i1
2 )3()1(
![Page 42: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/42.jpg)
Final Answer:
)3()1(...27123)3()1( 2
1
2 ni nn
i
i
![Page 43: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/43.jpg)
Series and Sigma Notation:
Properties of SumsSSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
![Page 44: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/44.jpg)
Property 1:
Property 1: The Sum of the First n Constant
cncn
i
1
![Page 45: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/45.jpg)
Proof:
cncn
i
1
n
n
i
ccccccc
...54321
1
![Page 46: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/46.jpg)
Property 2:
n
i
i
n
i
i acca11
![Page 47: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/47.jpg)
Proof:
n
i
i
n
i
i acca11
n
n
i
i cacacacacacaca
...54321
1
]...[ 54321
1
n
n
i
i aaaaaacca
![Page 48: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/48.jpg)
Property 3:Property 3: The Sum of the Sum or
Difference of Series
n
i
i
n
i
i
n
i
ii baba111
![Page 49: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/49.jpg)
Proof:
n
i
i
n
i
i
n
i
ii baba111
nn
n
i
ii bababababababa
...5544332211
1
nn
n
i
ii bbbbbbaaaaaaba
...... 5432154321
1
![Page 50: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/50.jpg)
Property 4:Property 4: The Sum of the First n
Natural Numbers
2
)1(
1
nni
n
i
![Page 51: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/51.jpg)
Property 5:Property 5: The Sum of the First n
Squares
6
)12)(1(
1
2
nnni
n
i
![Page 52: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/52.jpg)
Property 6:Property 6: The Sum of the First n
Cubes
2
1
3
2
)1(
nni
n
i
![Page 53: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/53.jpg)
Example 14:
Use the properties of sums to evaluate the series:
4
1
73i
i
![Page 54: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/54.jpg)
Final Answer:
2734
1
i
i
![Page 55: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/55.jpg)
Example 15:
Use the properties of sums to evaluate the series:
3
1
2 32i
ii
![Page 56: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/56.jpg)
Final Answer:
43323
1
2 i
ii
![Page 57: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/57.jpg)
Example 16:
Use the properties of sums to evaluate the series:
100
1
223
i
i
![Page 58: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/58.jpg)
Final Answer:
700,293,123100
1
2
i
i
![Page 59: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/59.jpg)
Example 17:
Write the following series in summation notation:
...20161284
![Page 60: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/60.jpg)
Final Answer:
n
i
i1
4...20161284
![Page 61: Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no last term](https://reader031.vdocument.in/reader031/viewer/2022012921/5e5f1341f7518516fd64796f/html5/thumbnails/61.jpg)
Performance Task 2:
Please download, print
and answer the “Let’s
Practice 22.” Kindly work
independently.