Sigma Notation and Partial Sums
Using Σ to represent the addition of terms within a sequence.
Introduction
Writing is tedious and imprecise. Sigma notation uses the Greek letter (sigma) to compactly represent sums. The notation specifies the starting index, ending index, and the pattern for each term—giving us a powerful shorthand for any sum.
Prerequisite Connection
You understand sequences and can find specific terms using formulas.
Today's Increment
We read and write sigma notation, and compute partial sums .
Why This Matters
Sigma notation is the language of series in calculus, statistics (expected values), and programming (loops). Mastery unlocks concise expression of complex sums.
Key Concepts
Sigma Notation Structure
Bottom: starting index
Top: ending index
Right: term formula
Partial Sum
The sum of the first terms:
Common Formulas
Properties of Summation
(constant factor)
(sum rule)
Worked Examples
Example 1: Expanding Sigma Notation (Basic)
Expand and evaluate
Write out each term (k = 1, 2, 3, 4):
Evaluate:
Example 2: Writing in Sigma Notation (Intermediate)
Express in sigma notation
Identify the pattern:
Each term is where
Answer:
Example 3: Using Formulas (Advanced)
Evaluate
Apply the formula :
Simplify:
Gauss's famous result!
Common Pitfalls
Off-by-one errors
has 5 terms, but has 6 terms! Count carefully.
Forgetting the index variable
The variable under Σ (like or ) is a "dummy variable"—it must appear in the formula.
Misapplying constant rule
(not ). You add a total of times.
Real-World Application
Expected Value in Statistics
In probability, the expected value of a random variable is written using sigma notation:
This compact notation expresses the weighted average of all possible outcomes—fundamental to statistics and decision theory.
Practice Quiz
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