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Sigma Notation
SIGMA NOTATION
Course
Mathematics For Engineers 1A (MAM1020F)
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Students shared 181 documents in this course
University
University of Cape Town
Academic year: 2023/2024
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Sigma Notation
Course: Mathematics For Engineers 1A (MAM1020F)
181 Documents
Students shared 181 documents in this course
University: University of Cape Town
Was this document helpful?
Sigma Notation
Consider a pattern of numbers. We call this a sequence. For example, 2, 4, 6, 8,…
Sums that have a pattern are called a series. For example, 2+4+6+8+…
A finite series is a sum that contains a fixed number of terms. We write it with the Greek letter Σ,
using sigma notation;
where 𝑚 and 𝑛 are integers and 𝑚 ≤ 𝑛.
Example 1:
i) ∑𝑖2
6
𝑖=1 = 12+ 22+ 32+ 42+ 52+ 62=91
ii) ∑𝑖
6
𝑖=1 = 1 + 2 + 3 + 4 + 5 + 6 = 20
iii) ∑1
𝑛
𝑖=1 = 1 + 1 + 1 + 1 + 1 + ⋯+ 1 = 𝑛
iv) ∑2𝑘
𝑛
𝑘=0 = 20+ 21+ 22+ 23+ 24+ ⋯+ 2𝑛
Example 2: Write the following in sigma notation.
i) 1 + 1
4+1
9+1
16
Solution: ∑1
𝑖2
4
𝑖=1
ii) 53+ 63+ 73+ 83+ ⋯+ 𝑛3
Solution: ∑𝑖3
𝑛
𝑖=5 𝑜𝑟 ∑(𝑖 + 3)3
𝑛−3
𝑖=2 (there are many ways to write the sum).
Some rules we need to follow. You can add and subtract sums or multiply them by a constant:
a) ∑𝑐𝑎𝑖=
𝑛
𝑖=𝑚 𝑐 ∑𝑐𝑎𝑖
𝑛
𝑖=𝑚
b) ∑(𝑎𝑖± 𝑏𝑖) =
𝑛
𝑖=𝑚 ∑𝑎𝑖±
𝑛
𝑖=𝑚 ∑𝑏𝑖
𝑛
𝑖=𝑚
Some useful formulae:
i) ∑1
𝑛
𝑖=1 = 𝑛
ii) ∑𝑐
𝑛
𝑖=1 =𝑐𝑛
iii) ∑𝑖
𝑛
𝑖=1 =𝑛(𝑛+1)
2
iv) ∑𝑖2
𝑛
𝑖=1 =𝑛(𝑛+1)(2𝑛+1)
6
v) ∑𝑖3
𝑛
𝑖=1 = (𝑛(𝑛+1)
2)2
