이항급수와 이항정리
이 항목의 스프링노트 원문주소
개요
\((1 + x)^\alpha = \sum_{k=0}^{\infty} {\alpha \choose k} x^k = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!} x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!} x^3 +\cdots\)
\(\frac{1}{(1-z)^a}=\sum_{n=0}^{\infty}\frac{(a)_n}{n!}z^n=1+az+\frac{a(a+1)}{2!}z^2+\frac{a(a+1)(a+2)}{3!}z^3+\cdots = \,_1F_0(a;z)\)
- 여기서 \((a)_n=a(a+1)(a+2)...(a+n-1)\)
예
\(\sqrt{1+x}=\sum _{k=0}^{\infty } \binom{\frac{1}{2}}{k} x^k=1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5 x^4}{128}+\frac{7 x^5}{256}-\frac{21 x^6}{1024}+\frac{33 x^7}{2048}-\frac{429 x^8}{32768}+\frac{715 x^9}{65536}-\frac{2431 x^{10}}{262144}+\cdots\)
\(\frac{1}{\sqrt{1-4z}}=\sum_{n=0}^{\infty} {{2n}\choose {n}} z^n=1+2 z+6 z^2+20 z^3+70 z^4+252 z^5+924 z^6+3432 z^7+12870 z^8+48620 z^9+184756 z^{10}+\cdots\)
역사
메모
관련된 항목들
매스매티카 파일 및 계산 리소스
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxYzg0NjZhOWMtOGUxNC00YjdkLTgxMTQtN2ExM2Y2NmIzZmNl&sort=name&layout=list&num=50
- http://www.wolframalpha.com/input/?i=
- http://functions.wolfram.com/
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
- Numbers, constants and computation
수학용어번역
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- [1]http://en.wikipedia.org/wiki/binomial_theorem
- http://en.wikipedia.org/wiki/Binomial_series
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
관련논문