Lemmas

This document introduces definitions, lemmas, and tricks that are universally used in mathematics and are difficult to categorize into specific categories.

Definitions

• Pi $\pi$
• Interval $[a,b]$
• Difference between a root and a solution
• Fixed point $x_{0} = f \left( x_{0} \right)$
• Harmonic mean
• Meaning of weak and strong in mathematics
• Closed form
• 해밍 거리 $H$
• Recurrence relation
• Continued fraction

Theorems

• Multiplication Formula Table
• Quickly Rationalizing Fractions Containing Roots
• Easily Squaring Two-Digit Numbers Ending in 5
• Expansion Formula for Double Radicals $\sqrt{a + b \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}$
• Quickly Finding the Extremes of a Quadratic Function
• Calculating the Sum of Square Numbers
• Calculating the Sum of an Arithmetic Series
  • Partial Sums of an Arithmetic Series are also Arithmetic
• Calculating the Sum of a Geometric Series
  • Partial Sums of a Geometric Series are also Geometric
• General Term of the Fibonacci Sequence
• Binomial Theorem
• Gabi’s Theorem
• Number of Subsets of a Finite Set with n Elements

Tricks

• How to move Big O notation from the denominator to the numerator
• Product of indicator functions 🔵
• Cumulative average formula 🔵

Inequalities

• Inequality between arithmetic, geometric, and harmonic means
• Cauchy-Schwarz inequality
  • Cauchy-Schwarz inequality for random variables 🔵
• Young’s inequality
• Bernoulli’s inequality
• Jensen’s inequality in finite form
• Jensen’s inequality in integral form
• Jensen’s inequality in expectation form 🔵
• Markov’s inequality 🔵
• Chebyshev’s inequality 🔵
• Grönwall’s inequality
• Inequality summarizing several inequalities 🔵
• Inequality for the logarithm function $1 - \dfrac{1}{x} \le \ln x \le x - 1$

Integral Formulas

• Definite integral of $e^{-x^{2}}$, Gaussian integral, Euler-Poisson integral
  • Generalization of the Gaussian integral
• Indefinite integral of $e^{x^{2}}$
• Table of integrals of trigonometric functions
• Integrability of $\dfrac{1}{x^p}$
• $\displaystyle \int_{-\infty}^{\infty} \dfrac{1}{1+x^{2}}dx = \pi$, $\displaystyle \int \dfrac{1}{1+x^{2}}dx = \tan^{-1} x + C$
• Frequently Used Definite Integrals of Quadratic Functions $\int_{\alpha}^{\beta}{(x-\alpha)(x-\beta)dx}=-\frac{{(\beta-\alpha)}^3}{6}$
• Integration of Natural Logarithms Raised to a Power $\int {(\ln x)^n} dx=x{(\ln x)^n}-\int n{(\ln x)^{n-1}}dx$

극한

• $\lim \limits_{x \to 0} \dfrac{\sin x}{x} = 1$
• $\lim \limits_{x \to 0} \dfrac{1 - \cos x}{x} = 0$
• $\lim\limits_{n \to \infty} \sqrt[n]{n} = 1$, $\lim\limits_{n \to \infty} \sqrt[n]{\dfrac{1}{n}} = 1$