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Mathematics Dictionary
Dr. K. G. Shih
The Story of Pi
Questions


  • Q01 | - arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ....
  • Q02 | - (pi)^2 = 6*(1 + 1/(2^2) + 1/(3^2) + ......)
  • Q03 | - (pi)^2 = 8*(1 + 1/(3^2) + 1/(5^2) + ......)
  • Q04 | - Archimedes polygon method to find pi
  • Q05 | - Machine-like formula
  • Q06 | -
  • Q07 | -
  • Q08 | -
  • Q09 | -
  • Q10 | -
    • Answers


    Q01. arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ....

    Series of pi
    • Since arctan(1) = pi/4
    • Hence pi/4 = 1 - 1/3 + 1/5 - 1/7 + .....
    • This series is simple but it coverge very slow

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    Q02. (pi)^2 = 6*(1 + 1/(2^2) + 1/(3^2) + 1/(4^2) + .....

    Keywords
    • 1. To prove it, we use series of sin(x)
    • 2. We also need equation of theory
    Proof
    • Sin(x) = x - (x^3)/(3!) + (x^5)/(5!)- (x^7)/(7!) + ....
    • If sin(x) = 0, the roots are r0 = 0, r1 = pi, r2 = 2*pi, r3 = 3*pi, ....
    • Hence r1, r2, r3, ... are roots of 1 - (x^2)/6 + (x^4)/120 - ..... = 0
    • Let x^2 = u,
        Then the equation becomes 1 - u/6 + (u^2)/120 - .... = 0
      • The roots of equation : u1 = (pi)^2, u2 = (2*pi)^2, u3 = (3*pi)^2
    • Equation theory
      • Equation : a*x^n + b*x^(n-1) + ..... + p*x + q = 0
      • Roots are x1, x2, x3, ....
      • Coefficient of x and roots : 1/x1 + 1/x2 + 1/x3 + ..... = p/q
    • Hence coefficient and roots u
      • 1/u1 + 1/u2 + 1/u3 + ..... = -(-1/6)/1
      • 1/(pi^2) + 1/((2*pi)^2) + 1/((3*pi)^2) + 1/((4*pi)^2) + .... = 1/6
      • Multiply (pi)^2 on both sides
      • 1 + 1/(2^2) + 1/(3^2) + ..... = ((pi)^2)/6

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    Q03. (pi)^2 = 8*(1 + 1/(3^2) + 1/(5^2) + 1/(7^2) + ....

    Keywords
    • 1. To prove it, we use series of cos(x)
    • 2. We also need equation of theory
    Proof
    • cos(x) = 1 - (x^2)/(2!) + (x^4)/(4!)- (x^6)/(6!) + ....
    • If cos(x) = 0, the roots are r1 = pi/2, r2 = 3*pi/2, r3 = 5*pi/2, ....
    • Hence r1, r2, r3, ... are roots of 1 - (x^2)/2 + (x^4)/24 - ..... = 0
    • Let x^2 = u,
        Then the equation becomes 1 - u/2 + (u^2)/24 - .... = 0
      • The roots of equation : u1 = (pi/2)^2, u2 = (3*pi/2)^2, u3 = (5*pi/2)^2
    • Equation theory
      • Equation : a*x^n + b*x^(n-1) + ..... + p*x + q = 0
      • Roots are x1, x2, x3, ....
      • Coefficient of x and roots : 1/x1 + 1/x2 + 1/x3 + ..... = p/q
    • Hence coefficient and roots u
      • 1/u1 + 1/u2 + 1/u3 + ..... = -(-1/2)/1
      • 1/((pi/2)^2) + 1/((3*pi/2)^2) + 1/((5*pi/2)^2) + .... = 1/2
      • Multiply (pi/2)^2 on both sides
      • 1 + 1/(3^2) + 1/(5^2) + ..... = ((pi/2)^2)/2 = ((pi)^2)/8

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    Q04. Archimedes polygon method to find pi

    Formula based Archimedes polygon method
    • pi = C*sin(A)
    • Where C = (2^n)/2 and A = 360/(2^n)
    • Where 2^n is the number of sides of polygon
    Example 1 : n = 3
    • pi = ((2^3)/2)*sin(45) = 4*Sqr(2)/2 = 2*Sqr(2)
    Example 1 : n = 5 and use half angle formula
    • pi = ((2^5)/2)*sin(360/32)
    • = 16*Sqr((1 - cos(360/16))/2)
    • = (16/Sqr(2))*Sqr(1 - cos(360/16))
    • = (16/Sqr(2))*Sqr(1 - Sqr(1 + cos(360/8))/2))
    • = 8*Sqr(2 - Sqr(2 + Sqr(2))) = 3.121445

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    Q05. Machine-like formula

    Formula
    • 1. Euler's formula : pi/4 = arctan(1/2) + arctan(1/3)
    • 2. Hermann's formula : pi/4 = 2*arctan(1/3) - arctan(1/7)
    • 3. Hutton's formula : pi/4 = 2*arctan(1/3) + arctan(1/7)

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    Q06. Answer

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    Q07. Answer

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    Q08. Answer

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    Q09. Answer

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    Q10. Answer

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