Pi

Math]Pi

 

数学知识忘地太快，在博客记录一下pi的生成。

• 100 Decimal places
  • 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
• Approximations
  • 22/7 3 decimal places (used by Egyptians around 1000BC)
  • 666/212 4 decimal places
  • 355/113 6 decimal places
  • 104348/33215 8 decimal places
• Series Expansions
  • English mathematician John Wallis in 1655.

       4 * 4 * 6 * 6 * 8 * 8 * 10 * 10 * 12 * 12 .....

  pi = 8 * -------------------------------------------------

       3 * 3 * 5 * 5 * 7 * 7 * 9 * 9 * 11 * 11 ....

  • Scottish mathematician and astronomer James Gregory in 1671

  pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ....)

  • Swiss mathematician Leonard Euler.

  pi = sqrt(12 - (12/22) + (12/32) - (12/42) + (12/52) .... )

下面则试证一下 Gregory’s Series

1. Taylor series

f(x)=∑n=0∞f(n)(a)n!(x−a)n(1)

2. Maclaurin series

f(x)=∑n=0∞f(n)(0)n!xn(2)

3. arctan(x)一阶导数

y=f(x)=arctan(x)x=tan(y)

⟹dxf′(x)=sec2y∗dy=dxdy=1x2+1

4. 推导过程

(1).y=arctan(x)的n阶导可以用下面的方法求得：

∵∴arctan(x)=∫x011+t2dt11+x2=12(11−ix+11+ix)arctan(x)=12i[ln(1−ix)−ln(1+ix)]

(2).若按原始方法，得先记住分数函数的求导方式：

(f(x)g(x))′=f′(x)g(x)−f(x)g′(x)g2(x)

(3).f(x)的n阶导数

 f(1)(x)=1x2+1f(2)(x)=−2x(x2+1)2f(3)(x)=2(3x2−1)(x2+1)3f(4)(x)=−24x(x2−1)(x2+1)4f(5)(x)=24(5x4−10x2+1)(x2+1)5...f(n)(x)=12(−1)ni[(−i+x)−n−(i+x)−n](n−1)!...

(4).f(x) Taylor Series Expansion 的系数

k1k2k3k4k5=f(1)(0)1!=1=f(2)(0)2!=0=f(3)(0)3!=−13=f(4)(0)4!=0=f(5)(0)5!=15...

5. get the conclusion, Maclaurin Series.

『Gregory's series』 or 『Leibniz's series』

∵arctan(x)∴arctan(1)=∑n=0∞(−1)n12n+1x2n+1=x−13x3+15x5−17x7+...=1−13+15−17+19−111+...=π4