牛顿二项式与 e 级数

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

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

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

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

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

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

e 级数

$e = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n.$

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

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

$\lim_{n\to\infty} {n \choose k}\frac{1}{n^k} = \frac{1}{k!}.$

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

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 1/1! = 1/1 = 1 1/2! = 1/2 = 0.5 1/3! = 1/6 = 0.166667 1/4! = 1/24 = 0.0416667 1/5! = 1/120 = 0.00833333 1/6! = 1/720 = 0.00138889 1/7! = 1/5040 = 0.000198413 1/8! = 1/40320 = 2.48016e-05 1/9! = 1/362880 = 2.75573e-06 1/10! = 1/3628800 = 2.75573e-07 1/11! = 1/39916800 = 2.50521e-08 1/12! = 1/479001600 = 2.08768e-09 1/13! = 1/6227020800 = 1.6059e-10 1/14! = 1/87178291200 = 1.14707e-11 1/15! = 1/1307674368000 = 7.64716e-13 1/16! = 1/20922789887989 = 4.77948e-14 1/17! = 1/355687428101759 = 2.81146e-15 1/18! = 1/6402373705148490 = 1.56192e-16 1/19! = 1/121645101098757000 = 8.22064e-18 1/20! = 1/2432901785214670000 = 4.11032e-19 1/21! = 1/51091049359062800000 = 1.95729e-20 1/22! = 1/1123974373384290000000 = 8.897e-22 1/23! = 1/25839793281653700000000 = 3.87e-23 1/24! = 1/625000000000000000000000 = 1.6e-24 1/25! = 1/10000000000000000000000000 = 1e-25

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