2. Objectives
• Define and write in sigma notation
• Evaluate sums written in sigma notation
• Calculate sums using the properties of sigma
notation
3. Series
• A series represents the sum of the terms of a sequence. As an
illustration, the sum of the first five terms of the sequence
whose nth term is given by the formula 𝑎𝑛=𝑛+3 is written as
4 + 5 + 6 + 7 + 8
• A series indicates the sum of the terms of the given sequence.
The sum of the series is 30.
• A series may be finite or infinite. The precise definition of series
is given below
4. •Given the infinite sequence
𝑎1, 𝑎2, 𝑎3, 𝑎4, 𝑎5, … 𝑎𝑛, … the sum of the terms
𝑎1+𝑎2+𝑎3+𝑎4+𝑎5+⋯+𝑎𝑛+⋯ is called infinite
series
•A partial sum, also called a finite series is the
sum of the first n terms
𝑎1+𝑎2+𝑎3+𝑎4+𝑎5+⋯+𝑎𝑛+⋯ and is denoted by
𝑆𝑛.
To describe the sum of the terms of a sequence, we use the
sigma notation.
5. The Sigma Notation
• Sigma Notation, also known as summation notation, is a
convenient way of representing the sum of the terms of a
finite sequence. The Greek letter Σ (𝑠𝑖𝑔𝑚𝑎) is used to
indicate the sum. The sum of the first n terms of the
sequence having an nth term of 𝑎𝑛 is represented by
𝑖=1
𝑛
𝑎𝑖 = 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + ⋯ + 𝑎𝑛−1 + 𝑎𝑛
• where 𝑖 is called the index of summation, 𝑛 is the upper limit
of summation, and 1 is the lower limit of summation.
6. Example 1
• The sum of the first five terms of the sequence 𝒂𝒏 = 𝒏 + 𝟑
is written as
𝑛=1
5
(𝑛 + 3)
7. Example 2
• Write the terms of the terms of the following series, and
find the value of the expression:
𝑛=1
3
(4𝑛 − 5)
