Chapter 1 Infinite Series. Definition of the Limit of a Sequence.
Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose...
Transcript of Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose...
Chapter 6:
Series and SequencesSSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Chapter 6: Series and Sequences
Lecture 22: Series
Lecture 23: Sequences
Lecture 21: Sequences
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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.
Two Kinds of Number Sequence
1. Finite Sequence; and2. Infinite Sequence.
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.
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.
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.
Example 1:
Finite or Infinite?
,...36,25,16,9,4,1
Example 2:
Finite or Infinite?
7,6,5,4,3,2,1
Example 3:
Finite or Infinite?
,...15,12,9,6,3
Example 4:
Finite or Infinite?
14,12,10,8,6,4,2
Example 5:
Finite or Infinite?
15,13,11,9,7,5,3,1
Example 6:
Finite or Infinite?
,...13,8,5,3,2,1,1
Example 7:
Write the first five terms of the sequence whose nth term is
given by the formula:
.23 nan
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
Example 8:
List the four terms of the infinite sequence whose nth
term is given by:
.)3()1( 2 nc n
n
Final Answer:
The first four terms of the sequence are
.1,0,1,4
Example 9:
Find the 9th term and the 21st
term of the sequence whose nth
term is:
.12
2
n
nan
Final Answer:
The 9th term is 81/80and the 21st term is
441/440.
Performance Task 1:
Please download, print
and answer the “Let’s
Practice 21.” Kindly work
independently.
Lecture 22: SeriesSSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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
Example 10:
Find S5 where:
.423 nan
Final Answer:
S5 is 55.
Example 11:
Find S4 where :
.)1( 2na n
n
Final Answer:
S4 is 10.
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.
Two Kinds of Series
1. Finite Series; and2.Infinite Series
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.
Finite Series
It is denoted by:
nn
n
i
i aaaaaaa
14321
1
...
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.
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.
Something to think about…
What is the correct way of reading the summation
notation? Let’s say
n
i
ia1
Did you know?
is read as “the summation of ai where i
is from 1 to n”.
n
i
ia1
Infinite Series
The sum of all terms of the infinite sequence is called a infinite series.
Infinite Series
It is denoted by:
nn
i
i aaaaaaa
14321
1
...
Take Note:
The upper limit summation of the finite series is n,
while in infinite series is ∞(symbol for infinity).
Example 12:
Write out the terms of the following series:
5
1
22i
i
Final Answer:
11025
1
2 i
i
Example 13:
Write out the terms of the following series:
n
i
i i1
2 )3()1(
Final Answer:
)3()1(...27123)3()1( 2
1
2 ni nn
i
i
Series and Sigma Notation:
Properties of SumsSSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Property 1:
Property 1: The Sum of the First n Constant
cncn
i
1
Proof:
cncn
i
1
n
n
i
ccccccc
...54321
1
Property 2:
n
i
i
n
i
i acca11
Proof:
n
i
i
n
i
i acca11
n
n
i
i cacacacacacaca
...54321
1
]...[ 54321
1
n
n
i
i aaaaaacca
Property 3:Property 3: The Sum of the Sum or
Difference of Series
n
i
i
n
i
i
n
i
ii baba111
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
Property 4:Property 4: The Sum of the First n
Natural Numbers
2
)1(
1
nni
n
i
Property 5:Property 5: The Sum of the First n
Squares
6
)12)(1(
1
2
nnni
n
i
Property 6:Property 6: The Sum of the First n
Cubes
2
1
3
2
)1(
nni
n
i
Example 14:
Use the properties of sums to evaluate the series:
4
1
73i
i
Final Answer:
2734
1
i
i
Example 15:
Use the properties of sums to evaluate the series:
3
1
2 32i
ii
Final Answer:
43323
1
2 i
ii
Example 16:
Use the properties of sums to evaluate the series:
100
1
223
i
i
Final Answer:
700,293,123100
1
2
i
i
Example 17:
Write the following series in summation notation:
...20161284
Final Answer:
n
i
i1
4...20161284
Performance Task 2:
Please download, print
and answer the “Let’s
Practice 22.” Kindly work
independently.