April 22, 2010

Sum[ (-1)^k*t^(2k),{k,0,Infinity}]

Integrate[1/(1+t^2),{t,0,x}]

Integrate[ Sum[ (-1)^k*t^(2k),{k,0,Infinity}],{t,0,x}]

Series[ ArcTan[x],{x,0,25}]

h[x_] = Normal[ Series[ ArcTan[x],{x,0,25}] ]

h[.3]

ArcTan[.3]

?Series

f[x_] = Normal[ Series[ ArcTan[x], {x,0,140}] ]

John Machin (1706)   (π/4) = 4ArcTan((1/5))  - ArcTan((1/239))

Tan[Pi/4]

Tan[ 4ArcTan[1/5]-ArcTan[1/239] ]   //  FullSimplify

N[4(4f[1/5]-f[1/239]),100]

N[Pi,100]

Euler    π = 20 ArcTan((1/7)) + 8 ArcTan((3/79))

Tan[Pi]

Tan[ 20ArcTan[1/7] + 8 ArcTan[3/79] ]   //  FullSimplify

f[x_] = Normal[ Series[ ArcTan[x], {x,0,120}] ];

N[ 20f[1/7]+8f[3/79],100 ]

N[Pi,100]

Newton   (π/2) = (∑_(k=0)^∞)(((2^k)(((k!))^2))/((2k+1)!))

N[2Sum[2^k*(k!)^2/(2k+1)!,{k,0,100}],100]

N[2Sum[2^k*(k!)^2/(2k+1)!,{k,0,200}],100]

N[2Sum[2^k*(k!)^2/(2k+1)!,{k,0,300}],100]

N[2Sum[2^k*(k!)^2/(2k+1)!,{k,0,400}],100]

F. C. W. Stoermer (1896) (1982)   

(π/4) = 44ArcTan((1/57)) + 7ArcTan((1/239)) - 12ArcTan((1/682)) + 24ArcTan((1/12943))

f[x_] = Normal[ Series[ ArcTan[x], {x,0,70}] ];

N[ 4( 44f[1/57]+7f[1/239]-12f[1/682]+24f[1/12943]) ,100]

K. Takano (1982)   (π/4) = 12 ArcTan((1/49)) +32 ArcTan((1/57)) -5 ArcTan((1/239))  + 24ArcTan((1/110443))

f[x_] = Normal[ Series[ ArcTan[x], {x,0,60}] ];

N[ 4( 12f[1/49]+32f[1/57]-5f[1/239]+12f[1/110443]),100]

N[Pi,100]

N[ 3 +8/60+29/(60)^2+44/(60)^3  ,20 ]

3+1/7 //N

3+1/(7+1/15)

N[ (333/106)]

3+1/(7+1/(15+1/1))

N[(355/113),10]