Write a repeated operation as one expression indexed by a variable: sigma notation and the rules for manipulating it, product notation, factorials, and indexed set operations.
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The 9 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.
01
Sigma notation
12 questions
Learn to read and write sigma notation - the compact mathematical shorthand for expressing sums.
Covers
Anatomy of sigma notation
Expanding sigma notation
Writing sums in sigma notation
02
Evaluating sums
12 questions
Learn how to expand and evaluate sigma expressions by substituting each index value into the general term, including sums with powers and fractional terms.
Covers
Evaluating a sum by expanding
Sums with powers of the index
Sums with fractional terms
03
Properties of sums
12 questions
Learn the rules for pulling out constant factors and splitting sums of additions or subtractions into separate sigma expressions.
Covers
Constant factor rule
Sum of sums rule
Difference of sums rule
04
Common sum formulas
8 questions
Learn the closed-form formulas for the sum of the first integers and the sum of the first squares.
Covers
Sum of first n integers
Sum of first n squares
05
Simplifying sums
8 questions
Learn to evaluate the sum of a constant term and to evaluate compound sigma expressions with linear and quadratic terms by decomposing them and substituting the formulas for and .
Covers
Sum of a constant
Evaluating sums with linear and quadratic terms
06
Re-indexing sums
12 questions
Learn to re-index sigma expressions by shifting the starting value, match a sum to a named expression through re-indexing, and determine whether two sums are equal by re-indexing one to match the other.
Covers
Shifting the index of summation
Re-indexing to apply known formulas
Determining whether two sums are equal
07
Product notation
12 questions
Learn the product operator and how to read, expand, evaluate, and write expressions in product notation.
Covers
Anatomy of product notation
Expanding and evaluating products
Writing products in product notation
08
Factorials
12 questions
Learn how to define and evaluate factorials, apply the recursive property, and simplify expressions involving factorial fractions.
Covers
Definition of factorial
The recursive property
Simplifying factorial expressions
09
Indexed unions and intersections
12 questions
Learn how to express unions and intersections of indexed set families using compact notation, and expand and compute these operations for rule-defined collections of sets.
Covers
Indexed union notation
Indexed intersection notation
Expanding and computing indexed set operations
This unit is part of our Mathematical Foundations for ML learning path, which contains 83 lessons. Every one of them is drawn below.
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9 lessons in this unit
7 prerequisite lessons across the unit, counting every step back to the start
67 other lessons in the learning path - after this unit, or alongside it
Builds on: Exponents and roots, Set operations
Leads to: Integrals
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More in Mathematical foundations: Inequalities and absolute value, Quadratic equations, Set operations, Exponents and roots, Functions, Logic, Exponentials and logarithms, Trigonometry