List of Series Formulas Latex Code

rockingdingo 2023-01-28 #math #series #taylor #binomial

List of Series Formulas Latex Code

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In this blog, we will summarize the latex code for series formulas, including arithmetic and geometric progressions, convergence of series: the ratio test, Binomial expansion, Taylor and Maclaurin Series, Power Series with Real Variables e^{x},ln(1+x),sin(x),cos(x), Plane Wave Expansion, etc.

• 1. Series
• 1.1 Arithmetic and Geometric progressions
• 1.2 Convergence of series: the ratio test
• 1.3 Binomial Expansion
• 1.4 Taylor Series
• 1.5 Power Series with Real Variables e^{x},ln(1+x),sin(x),cos(x)
• 1.6 Plane Wave Expansion

1. Series

• 1.1 Arithmetic and Geometric progressions

  Equation

  
  
  

  Latex Code

              S_{n}=a+(a+d)+(a+2d)+...+[a+(n-1)d]=\frac{n}{2}[2a+(n-1)d] \\
              S_{n}=a+ar+ar^{2}+...+ar^{n-1}=a\frac{1-r^{n}}{1-r}
          

  Explanation

  • Arithmetic progressions: Arithmetic progressions is calculated as . The first number is a, and the consecutive series number follows the pattern of at the number at the (n-1) th position is .
  • Geometric progressions: Geometric progressions is calculated as . The first number is a, and the consecutive series number follows the pattern of with multiplier as r and at the the number at (n-1) th position is .

  Related Documents

  • Mathematical Formula Handbook

• 1.2 Convergence of series: the ratio test

  Equation

  
  
  

  Latex Code

              S_{n}=u_{1}+u_{2}+...+u_{n} \\
              \text{Converge AS  } n \rightarrow \infty \text{, If} \lim_{n \rightarrow \infty} |\frac{u_{n+1}}{u_{n}}| < 1
          

  Explanation

  • The series will converge as , if holds.

  Related Documents

  • Mathematical Formula Handbook

• 1.3 Binomial Expansion

  Equation

  
  
  

  Latex Code

              (1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+...+ C^{r}_{n}x^{r} +...+x^{n} \\
              C^{r}_{n}=\frac{n!}{r!(n-r)!}
          

  Explanation

  • If n is a positive integer the series terminates, the term is . is combination symbol, which denotes the number of different ways in which an unordered sample of r objects can be selected from a set of n objects without replacement.

  Related Documents

  • Mathematical Formula Handbook

• 1.4 Taylor Series

  Equation

  
  
  

  Latex Code

              y(x)=y(a+u)=y(a)+u\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{1}{2!}u^{2}\frac{\mathrm{d}^2 y}{\mathrm{d} x^{2}}+\frac{1}{3!}u^{3}\frac{\mathrm{d}^3 y}{\mathrm{d} x^{3}}+...+\frac{1}{n!}u^{n}\frac{\mathrm{d}^n y}{\mathrm{d} x^{n}} + ...
          

  Explanation

  • : Factorials of n
  • : Derivative of y over x evaluated at point x=a
  • If y(x) is well-behaved in the vicinity of x = a then it has a taylor series as above equation.

  Related Documents

  • Mathematical Formula Handbook

• 1.5 Power Series with Real Variables e^{x},ln(1+x),sin(x),cos(x)

  Equation

  
  
  

  Latex Code

              e^{x}=1+x+\frac{x^{2}}{2!}+...+\frac{x^{n}}{n!}+... \\
  \ln(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} + ... + (-1)^{n+1}\frac{x^{n}}{n!} +... \\
  \cos(x) = \frac{e^{ix}+e^{-ix}}{2}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+...\\
  \sin(x) = \frac{e^{ix}-e^{-ix}}{2i}=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+...
          

  Explanation

  • Taylor Series Expansion of
  • 
  • Taylor Series Expansion of
  • 
  • Taylor Series Expansion of
  • 
  • Taylor Series Expansion of
  • 

  Related Documents

  • Mathematical Formula Handbook

• 1.6 Plane Wave Expansion

  Equation

  
  
  

  Latex Code

              \exp(ikz)=\exp(ikr\cos \theta=\sum^{\infty}_{l=0}(2l+1)i^{l}j_{l}(kr)P_{l}(\cos \theta)
          

  Explanation

  • : Legendre polynomials;
  • : spherical Bessel functions as . is the Bessel function of order l.

  Related Documents

  • Mathematical Formula Handbook