牛顿二项式与 e 级数

复习一下数学, 找一下回忆.

先是从二项式平方开始:

$(a+b)^{2} = a^{2} + 2ab + b^2$

其实展开是这样的:

$(a+b)^{2} = aa + ab + ba + bb$

再看立方:

$(a+b)^{3} = a^3 + 3a^{2}b + 3ab^2 + b^3$
$(a+b)^{3} = aaa + aab + aba + baa + bba + bab + abb + bbb$

通过排列组合的方式标记, 于是:

$(a+b)^3 = {3 choose 0}a^3 b^0 + {3 choose 1} a^{2}b^1 + {3 choose 2} a^{1}b^2 + {3 choose 3} a^{0} b^{3}$

通过数学归纳法可以拓展:

$(a+b)^n = {n choose 0}a^n b^0 + {n choose 1}a^{n-1}b^1 + {n choose 2}a^{n-2}b^2 + cdots + {n choose n-1}a^1 b^{n-1} + {n choose n}a^0 b^n$

使用求和简写可得:

$(a+b)^n = sum_{k=0}^n {n choose k}a^{n-k}b^k = sum_{k=0}^n {n choose k}a^{k}b^{n-k}$

e 级数

数学常数 e (The Constant e – NDE/NDT Resource Center) 的定义爲下列极限值:

$e = lim_{n oinfty} left(1 + frac{1}{n} ight)^n.$

使用二项式定理能得出

$left(1 + frac{1}{n} ight)^n = 1 + {n choose 1}frac{1}{n} + {n choose 2}frac{1}{n^2} + {n choose 3}frac{1}{n^3} + cdots + {n choose n}frac{1}{n^n}.$

第 k 项之总和为

${n choose k}frac{1}{n^k} ;=; frac{1}{k!}cdotfrac{n(n-1)(n-2)cdots (n-k+1)}{n^k}$

因为 n → ∞，右边的表达式趋近1。因此

$lim_{n oinfty} {n choose k}frac{1}{n^k} = frac{1}{k!}.$

由于序列的极限可以相加, 所以 e 可以表示为:

$e = sum_{k=0}^inftyfrac{1}{k!}=frac{1}{0!} + frac{1}{1!} + frac{1}{2!} + frac{1}{3!} + cdots.$

计算情况:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …

As an example, here is the computation of e to 22 decimal places:

1/0!	=	1/1	=	1.0000000000000000000000000
1/1!	=	1/1	=	1.0000000000000000000000000
1/2!	=	1/2	=	0.5000000000000000000000000
1/3!	=	1/6	=	0.1666666666666666666666667
1/4!	=	1/24	=	0.0416666666666666666666667
1/5!	=	1/120	=	0.0083333333333333333333333
1/6!	=	1/720	=	0.0013888888888888888888889
1/7!	=	1/5040	=	0.0001984126984126984126984
1/8!	=	1/40320	=	0.0000248015873015873015873
1/9!	=	1/362880	=	0.0000027557319223985890653
1/10!	=	1/3628800	=	0.0000002755731922398589065
1/11!	=	1/39916800	=	0.0000000250521083854417188
1/12!	=	1/479001600	=	0.0000000020876756987868099
1/13!	=	1/6227020800	=	0.0000000001605904383682161
1/14!	=	1/87178291200	=	0.0000000000114707455977297
1/15!	=	1/1307674368000	=	0.0000000000007647163731820
1/16!	=	1/20922789887989	=	0.0000000000000477947733239
1/17!	=	1/355687428101759	=	0.0000000000000028114572543
1/18!	=	1/6402373705148490	=	0.0000000000000001561920697
1/19!	=	1/121645101098757000	=	0.0000000000000000082206352
1/20!	=	1/2432901785214670000	=	0.0000000000000000004110318
1/21!	=	1/51091049359062800000	=	0.0000000000000000000195729
1/22!	=	1/1123974373384290000000	=	0.0000000000000000000008897
1/23!	=	1/25839793281653700000000	=	0.0000000000000000000000387
1/24!	=	1/625000000000000000000000	=	0.0000000000000000000000016
1/25!	=	1/10000000000000000000000000	=	0.0000000000000000000000001

For more information on e, visit the the math forum at mathforum.org
The sum of the values in the right column is 2.7182818284590452353602875 which is “e.”

Reference: The mathforum.org