Sequence and Series: Types A to Z

The terms sequence and series sound very similar, but they are quite different. Basically: A sequence is a set of ordered numbers, like 1, 2, 3, …, A series is the sum of a set of numbers, like 1 + 2 + 3…. Sequence and Series: Contents: Sequence Definition Sequence Rules Types of Sequences: Arithmetic … Read more

Polynomial Function: Definition, Examples, Degrees

Contents: Polynomial Function Definition Domain and Range of a Polynomial Degrees: First degree polynomial Second degree polynomial Third degree polynomial (a.k.a. Cubic Function) Limits for Polynomial Functions Polynomial Sequences Extreme Values of a Polynomial Types of Polynomial Function: Bernstein Polynomials Chebyshev Polynomials Hermite Polynomials Homogeneous polynomial Orthogonal Polynomials Symmetric Polynomials Univariate Polynomial Zernike Polynomials Zero … Read more

Derivatives / Differential Calculus: Definitions, Rules

  Contents: The Basics What is a Derivative? Formula: Limit Definition of a Derivative TI-89 Examples Differentiation Notation Rules for Derivatives Derivative Tables (Quick Reference Guide to Common Derivatives Constant Factor Rule The Product Rule Chain Rule Functions with exponents (the power rule). The Quotient Rule Reciprocal Rule: Definition, Examples The Sum Rule Specific Functions … Read more

Riemann Zeta Function

Types of Functions > Contents: What is the Riemann Zeta Function? Hurwitz Zeta Function Dirichlet L-Function What is the Riemann Zeta Function? The Riemann Zeta Function is a generalized harmonic series, given by the formula: Where n and s are two real numbers. When n = 1, the zeta function is the same as the … Read more

Different Types of Calculus: Traditional to Unusual

Calculus Definitions > Different Types of Calculus: Contents: Traditional (Infinitesimal) Calculus Real Analysis Non-Newtonian Calculus Stochastic Calculus Propositional, Ricci, Finite, Lambda & Umbral Traditional (Infinitesimal) Calculus The traditional calculus you come across in AP- or single-variable Calculus is also known as infinitesimal calculus. “Calculus” originally meant accounting or reckoning, originating with the Latin name for … Read more

Series Convergence Tests

Series Convergence Tests Contents (Click to skip to that section): Series Convergence Tests in Alphabetical Order Abel’s Test Absolute Convergence Alternating Series Convergence Tests Deleting the first N Terms Dirichlet’s Test Direct Comparison Test Geometric Series Convergence Tests Integral Series Convergence Tests nth-Term Test for Divergence P series Ratio Test Root Test Taylor Series Convergence … Read more

Dirichlet Function & Dirichlet Eta Function (Alternating Zeta Function)

Types of Functions > Contents: Dirichlet Function Dirichlet Eta Function Dirichlet Function The Dirichlet function is defined as: It’s a good example of a function with unusual limit behavior. At first glance, you might think that the function looks fairly simple, because it appears to neatly oscillate between zero and one. However, despite this apparent … Read more

Periodic Function, Aperiodic: Definition, Examples

Types of Functions > Contents: Periodic Function Aperiodic Function Almost Periodic What is a Periodic Function? A periodic function repeats its values at set intervals, called periods. A “function” is just a type of equation where every input (e.g. the x-value) results in a unique output (e.g. the y-value). More formally, we say that this … Read more

Differential Equations: Examples, Solutions

Differential equations have a derivative in them. For example, dy/dx = 9x. In elementary algebra, you usually find a single number as a solution to an equation, like x = 12. But with differential equations, the solutions are functions. In other words, you have to find an unknown function (or set of functions), rather than … Read more