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! colspan="2" style="background: #e4bc7e; font-size: 110%;" | Power Series Formulas | ! colspan="2" style="background: #e4bc7e; font-size: 110%;" | Power Series Formulas | ||
|- | |- | ||
| − | ! colspan="2" style="background: #eee;" | Taylor Series | + | ! colspan="2" style="background: #eee;" | Series in symbolic forms |
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Taylor Series in one variable || <math> \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | Taylor Series in d variables || | ||
| + | <math>=\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} | ||
| + | \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d).\!</math> | ||
| + | |- | ||
| + | ! colspan="2" style="background: #eee;" | Taylor Series of certain functions | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | exponential || <math>e^x = \sum_{n=0}^\infty \frac{x^n}{n!},</math> for all <math> x\in {\mathbb C}\ </math> | | align="right" style="padding-right: 1em;" | exponential || <math>e^x = \sum_{n=0}^\infty \frac{x^n}{n!},</math> for all <math> x\in {\mathbb C}\ </math> | ||
Revision as of 16:42, 2 November 2009
| Power Series Formulas | |
|---|---|
| Series in symbolic forms | |
| Taylor Series in one variable | $ \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} $ |
| Taylor Series in d variables |
$ =\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d).\! $ |
| Taylor Series of certain functions | |
| exponential | $ e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, $ for all $ x\in {\mathbb C}\ $ |
| Geometric Series | |
| (info) Finite Geometric Series Formula | $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right. $ |
| (info) Infinite Geometric Series Formula | $ \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. $ |
| Other Series | |
| notes/name | equation |
