Sigma Notation Formula
The sigma notation formula \sum_{i=m}^{n} a_i = a_m + a_{m+1} + \cdots + a_n is shorthand for summing many terms. The index runs from m to n, and the expression to the right tells you what to add each step.
The Formula
When to use: Sigma notation is shorthand for 'add these all up.' The letter below \Sigma is a counter, the number below is where to start, the number above is where to stop, and the expression to the right tells you what to add each time.
Quick Example
Notation
What This Formula Means
Sigma notation uses the Greek letter ฮฃ to express the sum of many terms compactly. The expression \sum_{i=1}^{n} a_i means 'add up a_i for every integer i from 1 to n.' For example, \sum_{i=1}^{4} i^2 = 1 + 4 + 9 + 16 = 30.
Sigma notation is shorthand for 'add these all up.' The letter below \Sigma is a counter, the number below is where to start, the number above is where to stop, and the expression to the right tells you what to add each time.
Formal View
Worked Examples
Example 1
easySolution
- 1 Write out each term: k=1: 1, k=2: 3, k=3: 5, k=4: 7, k=5: 9.
- 2 Sum: 1+3+5+7+9 = 25.
- 3 Alternatively, use linearity: 2\sum_{k=1}^5 k - \sum_{k=1}^5 1 = 2 \cdot 15 - 5 = 25.
Answer
Example 2
mediumCommon Mistakes
- Misreading the bounds: \sum_{i=0}^{n} has n+1 terms (0, 1, ..., n), while \sum_{i=1}^{n} has n terms.
- Pulling non-constant terms outside the sum: \sum_{i=1}^{n} c \cdot a_i = c \sum_{i=1}^{n} a_i only works if c does not depend on i.
- Confusing \sum_{i=1}^{n} (a_i + b_i) = \sum a_i + \sum b_i (TRUE) with \sum_{i=1}^{n} a_i \cdot b_i = (\sum a_i)(\sum b_i) (FALSE).
Why This Formula Matters
Nearly every branch of mathematics uses summation notation: series in calculus, expected values in probability, dot products in linear algebra, and algorithms in computer science. Fluency with \Sigma is a prerequisite for higher math.
Frequently Asked Questions
What is the Sigma Notation formula?
Sigma notation uses the Greek letter ฮฃ to express the sum of many terms compactly. The expression \sum_{i=1}^{n} a_i means 'add up a_i for every integer i from 1 to n.' For example, \sum_{i=1}^{4} i^2 = 1 + 4 + 9 + 16 = 30.
How do you use the Sigma Notation formula?
Sigma notation is shorthand for 'add these all up.' The letter below \Sigma is a counter, the number below is where to start, the number above is where to stop, and the expression to the right tells you what to add each time.
What do the symbols mean in the Sigma Notation formula?
\sum_{i=1}^{n} a_i โ the index variable i is a dummy variable (can be replaced by j, k, etc.).
Why is the Sigma Notation formula important in Math?
Nearly every branch of mathematics uses summation notation: series in calculus, expected values in probability, dot products in linear algebra, and algorithms in computer science. Fluency with \Sigma is a prerequisite for higher math.
What do students get wrong about Sigma Notation?
The index variable is a dummyโ\sum_{i=1}^{n} i^2 and \sum_{k=1}^{n} k^2 are identical. The variable disappears after summation, just like a definite integral's variable.
What should I learn before the Sigma Notation formula?
Before studying the Sigma Notation formula, you should understand: sequence, series.