5_Week-5_ PRE CALCULUS SIGMA NOTATION Series.pdf
- 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)
- 8. Example 3 • Evaluate: 𝑛=1 5 −1 𝑛 2𝑛 𝑛 + 1
- 9. Example 4 • Write the series in sigma notation 2 + 4 + 6 + 8 + 10
- 10. Example 5 • Write the series in sigma notation 1 − 1 2 + 1 3 − 1 4 + ⋯ − 1 10