Lesson 19.2

Sigma Notation and Partial Sums

Using Σ to represent the addition of terms within a sequence.

Introduction

Using Σ to represent the addition of terms within a sequence.

Past Knowledge

You understand sequences and can find specific terms using formulas.

Today's Goal

We read and write sigma notation, and compute partial sums .

Future Success

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.