## List of Series Formulas Latex Code

rockingdingo 2023-01-28 #math #series #taylor #binomial 2 0

List of Series Formulas Latex Code

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

##### 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 .

• #### 1.2 Convergence of series: the ratio test

##### 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.

• #### 1.3 Binomial Expansion

##### 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.

• #### 1.4 Taylor Series

##### 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.

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

##### 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

• #### 1.6 Plane Wave Expansion

##### 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.