"टेलर श्रेणी" (अनुभाग) सम्पादित करें

== कुछ सामान्य फलनों के लिए मैक्लारिन श्रेणियाँ==
[[Image:TaylorCosCosEnhanced.svg|150px|thumb|right|The real part of the cosine function in the [[complex plane]]]]
[[Image:TaylorCosPolSVG.svg|150px|thumb|right|An 8th-degree approximation of the cosine function in the [[complex plane]]]]
[[Image:TaylorCosAllSVG.svg|150px|thumb|right|The two above curves put together]]
<!-- [[Image:TaylorCosAnim.gif|280px|thumb|right|An animation of the approximation.]] -->
नीचे बहुत से महत्वपूर्ण मैक्लारिन श्रेणी प्रसार दिए गए हैं।<ref>{{harv|Abramowitz|Stegun|1970}}.</ref> ये सभी प्रसार समिश्र अर्गुमेन्ट ''x'' के लिए सत्य हैं (अतः वास्तविक के लिए भी सत्य हैं।)।

'''[[चरघातांकी फलन]]''' (Exponential function)
:<math>e^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\quad\text{ for all } x\!</math>

'''[[प्राकृतिक लघुगणक]]''':
:<math>\log(1-x) = - \sum^{\infin}_{n=1} \frac{x^n}n\quad\text{ for } |x| < 1</math>

:<math>\log(1+x) = \sum^\infin_{n=1} (-1)^{n+1}\frac{x^n}n\quad\text{ for } |x| < 1</math>

'''[[गुणोत्तर श्रेणी]]''' :

:<math>\frac{1}{1-x} = \sum^\infin_{n=0} x^n\quad\text{ for }|x| < 1\!</math>

[[द्विपद श्रेणी]] (Binomial series) (includes the square root for ''α'' = 1/2 and the infinite geometric series for ''α'' = &minus;1):

:<math>(1+x)^\alpha = \sum_{n=0}^\infty {\alpha \choose n} x^n\quad\text{ for all }|x| < 1 \text{ and all complex } \alpha\!</math>

with generalized [[binomial coefficient]]s

: <math>{\alpha\choose n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}</math>

[[Trigonometric function]]s:

:<math>\sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\quad\text{ for all } x\!</math>

:<math>\cos x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\quad\text{ for all } x\!</math>

:<math>\tan x = \sum^{\infin}_{n=1} \frac{B_{2n} (-4)^n (1-4^n)}{(2n)!} x^{2n-1} = x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots\quad\text{ for }|x| < \frac{\pi}{2}\!</math>

:<math>\sec x = \sum^{\infin}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n}\quad\text{ for }|x| < \frac{\pi}{2}\!</math>

:<math>\arcsin x = \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\!</math>

:<math>\arccos x ={\pi\over 2}-\arcsin x={\pi\over 2}- \sum^{\infin}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\!</math>

:<math>\arctan x = \sum^{\infin}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1}\quad\text{ for }|x| \le 1, x\not=\pm i\!</math>

[[Hyperbolic function]]s:

:<math>\sinh x = \sum^{\infin}_{n=0} \frac{x^{2n+1}}{(2n+1)!} = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots\quad\text{ for all } x\!</math>

:<math>\cosh x = \sum^{\infin}_{n=0} \frac{x^{2n}}{(2n)!} = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots\quad\text{ for all } x\!</math>

:<math>\tanh x = \sum^{\infin}_{n=1} \frac{B_{2n} 4^n (4^n-1)}{(2n)!} x^{2n-1} = x-\frac{1}{3}x^3+\frac{2}{15}x^5-\frac{17}{315}x^7+\cdots \quad\text{ for }|x| < \frac{\pi}{2}\!</math>

:<math>\mathrm{arcsinh} (x) = \sum^{\infin}_{n=0} \frac{(-1)^n (2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1}\quad\text{ for }|x| \le 1\!</math>

:<math>\mathrm{arctanh} (x) = \sum^{\infin}_{n=0} \frac{x^{2n+1}}{2n+1} \quad\text{ for }|x| \le 1, x\not=\pm 1\!</math>

The numbers ''B''<sub>''k''</sub> appearing in the ''summation'' expansions of tan(''x'') and tanh(''x'') are the [[Bernoulli numbers]]. The ''E''<sub>''k''</sub> in the expansion of sec(''x'') are [[Euler number]]s.