Series and the
Sigma Notation
Mr. Christopher C. Dacanay
Senior High School
University of Santo Tomas
This material was prepared by the teacher for the exclusive and personal use of those students who are officially enrolled in the section. Some parts of the
presentation were lifted from various print and online resources. Any error that may be found in the material was not intentionally committed by the teacher.
Lord, true source of light and wisdom
Give me a keen sense of understanding
A retentive memory
And the capacity to grasp things correctly.
Grant me the grace
To be accurate in my expositions
And the skill to express myself
With thoroughness and clarity.
Be with me at the start of my work
Guide its progress
And bring it to completion.
Grant this through Christ, our Lord.
Amen.
Learning Competencies
• illustrate a series
• differentiate a series from a sequence
• use the sigma notation to represent a series
• apply the use of sigma notation in finding sums
PRE-CALCULUS: Series and The Sigma Notation
Sequence: A Brief Review
Recall that a sequence is a function whose domain is the set of positive
integers and whose range is the set of real numbers. For convenience, we denote
the value of the function at n by 𝑎𝑛 , which also represents the nth term of the
sequence. In this instance the sequence is denoted by 𝑎𝑛 .
Remark: Sequences may be finite or infinite.
In previous grades, you talked about arithmetic, harmonic, and geometric
sequences. Try to recall and answer the first example.
Example 1 Determine whether the given sequence is arithmetic, harmonic, or geometric.
Find the 7th term of each sequence.
PRE-CALCULUS: Series and The Sigma Notation
Series and the Sigma Notation
A series represents the sum of the terms of a sequence. Given a sequence 𝑎𝑛 ,
the expression
𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛
is called a series.
For convenience, we shall write the expression 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛 as
𝑛
𝑎𝑛
𝑘=1
where σ, is called the sigma notation.
Thus, at this point we can simply say that a sequence is a list of numbers,
while a series is a sum of numbers.
Remarks: 1. Similar to a sequence, a series may be finite or infinite.
2. In the sigma notation, the value of 𝑘 may not need to start at 1.
PRE-CALCULUS: Series and The Sigma Notation
Example 2 Write the corresponding series for the given sequence.
Example 3 Write the following in sigma notation.
PRE-CALCULUS: Series and The Sigma Notation
Example 4 Evaluate each series.
PRE-CALCULUS: Series and The Sigma Notation
Properties of Sigma Notation
PRE-CALCULUS: Series and The Sigma Notation
Example 5
PRE-CALCULUS: Series and The Sigma Notation
Some Summation Formulas
Example 6
Find the value of each sum.
PRE-CALCULUS: Series and The Sigma Notation
Some Summation Formulas
Example 7
Find the value of each sum.
PRE-CALCULUS: Series and The Sigma Notation
Practice Exercises
PRE-CALCULUS: Series and The Sigma Notation
CLOSING PRAYER
We give You thanks, Almighty God, for all the good things
You have given us today; for wisdom to understand, for
moral courage to avoid wrong and for the love to act kindly
towards others. Oh God, we are truly sorry for any thought,
word or deed by which we have offended Your divine
goodness or willfully wronged our neighbors.
Help us to be better persons everyday.
Amen.
