Citeprep Guides

Welcome to CitePrep Guides

Every web page on this CitePrep Guides comes with a context-sensitive help page. Simply click on the circle question mark located at the top right section of every page.

Click on

to expand a section.

Click on

to collapse a section.

Click on to collaps all lines on a page.

Click on to expand all lines on a page.

HELP

Click on the HELP icon to launch this help page.

CLEVELAND

Click on the PROFILE icon to check your current status.

GAMES

Click on the GAME icon to launch the panel on games

CARDS

Click on the CARDS icon to launch the panel on flashcards

TERMS

Click on the TERMS icon to launch the panel on key terms.

Click on the MENU icon to launch the panel on menus.

DAY

NIGHT

Reference

Scroll down to your desired entry,
noted in reverse colors.

Hand

Fish

Relax

A familiar exercise for anyone suffering from anxiety is to envision themselves in a safe place away from the triggers that induce their condition. These animations are meant to be a substitute for such visualizations.

Watching a fish in an aquarium or the five-finger breathing exercise is meant to help with anxiety. My goal here is to create a suite of animations that mitigate symptoms of anxiety. Future animations include fireflies flowing at night, waves on a seashore, a night sky, and random birds frequenting a barren tree.

Index

Temporary

Notes

General

Taylor polynomials

What are Taylor Polynomials
Why are Taylor Polynomials important
What is the history of Taylor Polynomials
How do you build a Taylor Polynomial
How do the graphs of Taylor Polynomials appear
Build a Taylor Polynomial for EACH function listed in Table I below
What is error bound for a Taylor Polynomial
Calculate the error bound for a Taylor Polynomial

Infinite sequences

What is a sequence
What is an infinite sequence
What are some famous sequences
Arithmetic Sequence
Geometric Sequence
What is a Series
Limit of a sequence

Limit of a sequence

Show convergence of a sequence as a continuous function
Show convergence by definition
Show convergence using the squeeze theorem
Show convergence for a geometric sequence
Show convergence by L'Hopital's Rule
Show convergence for alternating sequences
Show convergence by the Absolute Value Theorem
Finding the Nth Term of a sequence
Determine whether a sequence is monotonic
Determine whether a sequence is bounded
Apply limit laws for sequences

Power series

Review a Taylor Polynomial
What is a Power Series
What is the history of a Power Series?
How do you form a Power Series
The Geometric Series as a Power Series
Does a Power Series converge?

Infinite series

What is an infinite series?
When is an infinite series convergent?
When is an infinite series divergent?
What is a telescoping series?
When is a geometric series convergent?
Using the properties of an infinite series

Testing for convergence

Nth-Term Test for divergence
Integral Test for convergence
P-Series Test for convergence
Direct Comparison Test for convergence
Limit Comparison Test for convergence
Alternating Series Test for convergence
Alternating Series Remainder
Testing for absolute and conditional convergence
Ratio Test for convergence
Root Test for convergence
Strategies for testing convergence

Maclaurin Series

What is a Maclaurin Series (MS)?
What is the MS for the exponential function?
What is the MS for the sine function?
What is the MS for the cosine function?
What is the MS for the arctangent function?
What is the MS for the natural log function?
Synthesize new MSs using integration
Synthesize new MSs using derivatives
Synthesize new MSs using multiplication
Synthesize new MSs using substitution
Given any MS, find its radius of conversion

Taylor Series

What is a Taylor Series (TS)?
What is the TS for the exponential function at a=1?
What is the TS for the sine function at a=1?
What is the TS for the cosine function at a=1?
What is the TS for the arctangent function at a=1?
What is the TS for the natural log function at a=1?
Synthesize new TSs using integration at a=1
Synthesize new TSs using derivatives at a=1
Synthesize new TSs using multiplication at a=1
Synthesize new TSs using substitution at a=1
Given any TS, find its radius of conversion at a=1

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

Introduction

Settings

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Faucibus et molestie ac feugiat sed lectus vestibulum. Maecenas ultricies mi eget mauris pharetra et ultrices neque ornare. Id aliquet lectus proin nibh. Sollicitudin tempor id eu nisl. Mus mauris vitae ultricies leo integer malesuada nunc. Et netus et malesuada fames ac turpis egestas. Ultricies tristique nulla aliquet enim tortor at auctor. Egestas congue quisque egestas diam in. Bibendum est ultricies integer quis auctor. Ut ornare lectus sit amet est placerat in egestas. Pellentesque nec nam aliquam sem. Nunc mattis enim ut tellus elementum sagittis. Phasellus egestas tellus rutrum tellus pellentesque eu tincidunt tortor.

Games

Quizzes

Memory

Glossary ZZ

Wait for it

Theorems

Definitions

Vocabulary

History

1000's

1100's

1200's

1300's

1400's

1500's

1600's

1700's

1800's

1900's

2000's

2100's

Introduction of ZERO

1170-1180

Bubonic Plague

1665-1666

Napoleonic Wars

1803-1815

Leonardo Bigollo Pisano

80 years

1170-1250

John Napier

67 years

1550-1617

René Descartes

54 years

1596-1650

Isaac Newton

84 years

1643-1727

Colin Maclaurin

48 years

1698-1746

Leonhard Euler

76 years

1707-1783

Carl Friedrich Gauss

78 years

1777-1855

Augustin-Louis Cauchy

68 years

1789-1857

Ada Lovelace

37 years

1815-1852

Bernhard Riemann

40 years

1826-1866

Maryam Mirzakhani

40 years

1977-2017

Click on an Event or Name

Flashcards

PROVE:

\displaystyle \lim_{x\rightarrow \infty} \frac 1x = 0.

Example of arranging a sequence of steps in math

PROOF

\LARGE \begin{gather*} \lim_{x\rightarrow \infty} \frac 1x = 0 \\[1em] \textrm{if} \ \ \forall \epsilon \gt 0 \ \ \exists M \gt 0 \\[1em] \textrm{such that} \ \ \left| \frac 1x \right| \lt \epsilon \ \ \textrm{when} \ \ x \gt M.\\ \end{gather*}

Let $\epsilon \gt 0$ be chosen. We must find some $M \gt 0$ so that $\left| \frac 1x \right| \lt \epsilon$ when $x \gt M.$

Take $M = 1/\epsilon$ and assume $x \gt M.$ Now,

\LARGE \begin{array}{ll} x \gt M & \color{lightgrey}\textrm{by assumption},\\[1em] x \gt 1/\epsilon & \color{lightgrey}M= 1/\epsilon, \textrm{ as taken},\\[1em] |x| \gt 1/\epsilon & \color{lightgrey}x \rightarrow \infty \Rightarrow x \gt 0,\\[1em] \left| \frac 1x \right| \lt \epsilon & \color{lightgrey}\textrm{Reciprocal}.\\[1em] \qedsymbol \end{array}

$x \gt M = 1/\epsilon,$ $|x| \gt 1/\epsilon,$ and therefore $\left| \frac 1x \right| \lt \epsilon.$

, then choose $M = \frac 1\epsilon.$ Now, suppose $x \gt M,$ then $|x|\gt M,$ $|\frac 1x|\gt M,$

Note $\left| \frac 1x \right| \lt \epsilon,$ $| x | \gt \frac 1\epsilon,$ $x \gt \frac 1\epsilon,$ $x \gt M.$

\LARGE \begin{gather*} \lim_{x\rightarrow \infty} \frac 1x = 0 \ \ \textrm{ is true }\\[1em] \textrm{if} \ \ \forall \epsilon \gt 0 \ \ \exists M \gt 0 \\[1em] \textrm{such that} \ \ \left| \frac 1x \right| \lt \epsilon \ \ \textrm{when} \ \ M \gt x.\\ \end{gather*}