| Line 13: | Line 13: | ||
! colspan="2" style="background: #eee;" | Taylor Series of certain functions | ! 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> | + | | align="right" style="padding-right: 1em;" | exponential || <math>e^x = \sum_{n=0}^\infty \frac{x^n}{n!},</math> <math> \text{ for all } x\in {\mathbb C}\ </math> |
|- | |- | ||
| − | ! colspan="2" style="background: #eee;" | Geometric Series | + | | align="right" style="padding-right: 1em;" | logarithm || |
| + | <math>\ln(1+x) = \sum^{\infin}_{n=1} (-1)^{n+1}\frac{x^n}n,\text{ when }-1<x\le1</math> | ||
| + | |- | ||
| + | ! colspan="2" style="background: #eee;" | Geometric Series and related series | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | [[more_on_geometric_series|(info)]] Finite Geometric Series Formula || <math>\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. </math> | | align="right" style="padding-right: 1em;" | [[more_on_geometric_series|(info)]] Finite Geometric Series Formula || <math>\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. </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | [[more_on_geometric_series|(info)]] Infinite Geometric Series Formula || <math>\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. </math> | | align="right" style="padding-right: 1em;" | [[more_on_geometric_series|(info)]] Infinite Geometric Series Formula || <math>\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. </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | || <math>\frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for }|x| < 1 \text{ and } m\in\mathbb{N}_0\!</math> | ||
|- | |- | ||
! colspan="2" style="background: #eee;" | Other Series | ! colspan="2" style="background: #eee;" | Other Series | ||
Revision as of 16:51, 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!}, $ $ \text{ for all } x\in {\mathbb C}\ $ |
| logarithm |
$ \ln(1+x) = \sum^{\infin}_{n=1} (-1)^{n+1}\frac{x^n}n,\text{ when }-1<x\le1 $ |
| Geometric Series and related 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. $ |
| $ \frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n\quad\mbox{ for }|x| < 1 \text{ and } m\in\mathbb{N}_0\! $ | |
| Other Series | |
| notes/name | equation |
