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Mathematics Dictionary
Dr. K. G. Shih
Integral : Seires of functions
Subjects


  • CA 18 00 | - Outlines
  • CA 18 01 | - Series of arctan(x)
  • CA 18 02 | - Seires of ln(1 + x)
  • CA 18 03 | - Series of ln(1 - x)
  • CA 18 04 | - series of e^x
  • CA 18 05 | - Series of e^(-x)
  • CA 18 06 | - Series of sinh(x)
  • CA 18 07 | - series of cosh(x)
  • CA 18 08 | - Series of sin(x)
  • CA 18 09 | - Series of cos(x)
  • CA 18 10 | - Series of arcsin(x)
    • Answers


    CA 18 01. Series of arctan(x)

    Integral and arctan(x)
    • arctan(x) = ∫(1/(1 + x^2))dx
    Expand 1/(1+x^2) using binomial theory
    • 1/(1 + x^2)
    • = (1 + x^2)^(-1)
    • = 1 + (-1)*x^2 + (-1)*(-1-1)*((x^2)^2)/2! + (-1)*(-1-1)(-1-2)*((x^3)^3)/3! + ....
    • = 1 - x^2 + x^4 - x^6 + x^8 - ....
    Integral by power rule
    • arctan(x) = ∫(1/(1 + x^2)dx
    • = ∫(1 - x^2 + x^4 - x^6 + x^8 - ....)dx
    • = x - (x^3)/3 + (x^5)/5 - (x^7)/7 + ....
    pi = 4*arctan(1)
    • pi = 4*(1 - 1/3 + 1/5 - 1/7 + 1/9 - .....)
    • How many terms are required to get pi = 3.14 ?

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    CA 18 02. Series of ln(1 + x)

    Integral and ln(1 + x)
    • ln(1 + x) = ∫(1/(1 + x))dx
    Expand 1/(1+x) using binomial theory
    • 1/(1 + x)
    • = (1 + x)^(-1)
    • = 1 + (-1)*x + (-1)*(-1-1)*((x)^2)/2! + (-1)*(-1-1)(-1-2)*((x)^3)/3! + ....
    • = 1 - x + x^2 - x^3 + x^4 - ....
    Integral by power rule
    • ln(1 + x) = ∫(1/(1 + x))dx
    • = ∫(1 - x + x^2 - x^3 + x^4 - ....)dx
    • = x - (x^2)/2 + (x^3)/3 - (x^4)/4 + ....
    Find ln(2)
    • Let x = 1
    • ln(2) = 1 - 1 + 1/2 - 1/3 + 1/4 - 1/5 + 1/6 - 1/7 .....
    • = 0.5 - 0.33333 + 0.25 - 0.2 + 0.166667 - .....

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    CA 18 03. Series of ln(1 - x)

    Integral and ln(1 - x)
    • ln(1 - x) = ∫(-1/(1 - x))dx
    Expand 1/(1-x) using binomial theory
    • 1/(1 - x)
    • = (1 - x)^(-1)
    • = 1 + (-1)*(-x) + (-1)*(-1-1)*((-x)^2)/2! + (-1)*(-1-1)(-1-2)*((-x)^3)/3! + ....
    • = 1 + x + x^2 + x^3 + x^4 - ....
    Integral by power rule
    • ln(1 - x) = ∫(-1/(1 - x))dx
    • = ∫-(1 + x + x^2 + x^3 + x^4 - ....)dx
    • = -x - (x^2)/2 - (x^3)/3 - (x^4)/4 - ....
    Find ln(0.9)
    • Let x = 0.1
    • ln(1 - 0.1) = -(0.1 + ((0.1)^2)/2 + ((0.1)^3)/3 + ((0.1)^4)/4 - .....)
    • ln(0.9) = -(1 + 0.1 + 0.005 + 0.00033 + 0.000025 + ....)
    • ln(0.9) = -0.10536



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    CA 18 04. Series of e^x

    Taylor theory
    • F(x) = D0(0) + D1(0)*x + D2(0)*(x^2)/2! + D3(0)*(x^3)/3! + .....
    • D0(0) is the F(0)
    • D1(0) is 1st derivative of F(x) = F'(0)
    • D2(0) is 2nd derivative of F(x) = F"(0)
    • D3(0) is 3rd derivative of F(x) at x = 0
    • D4(0) is 4th derivative of F(x) at x = 0
    Use taylor theory
    • Since the derivative of e^x is e^x, hence
    • D0(0) = e^(0) = 1
    • D1(0) = e^(0) = 1
    • D2(0) = e^(0) = 1
    • D3(0) = e^(0) = 1
    • Hence e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ......
    Find e^1
    • Let x = 1
    • e = 1 + 1 + 1/2! + 1/3! + 1/4! + ....
    • e = 2 + 0.5 + 0.166666 + 0.0416666 + ....
    • e = 2.71828....

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    CA 18 05. Seires of e^(-x)

    Use taylor theory
    • Since the derivative of e^(-x) is -e^(-x), hence
    • D0(0) = +e^(0) = 1
    • D1(0) = -e^(0) = 1
    • D2(0) = +e^(0) = 1
    • D3(0) = -e^(0) = 1
    • Hence e^(-x) = 1 - x + (x^2)/2! - (x^3)/3! + (x^4)/4! - ......

    Go to Begin

    CA 16 06. Seires of sinh(x)

    Use taylor theory
    • Since the derivative of sinh(x) is cosh(x) and derivative cosh(x) = sinh(x) hence
    • D0(0) = sinh(0) = 0
    • D1(0) = cosh(0) = 1
    • D2(0) = sinh(0) = 0
    • D3(0) = cosh(0) = 1
    • Hence sinh(x) = 0 + x + 0 + (x^3)/3! + 0 + (x^5)/5! +......
    • Hence sinh(x) = x + (x^3)/3! + (x^5)/5! + ....
    2nd method : cosh(x) = (e^(+x) - e(-x))/2
    • e^(+x) = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ......
    • e^(-x) = 1 - x + (x^2)/2! - (x^3)/3! + (x^4)/4! + ......
    • sinh(x) = ((e^(+x) - e^(-x))/2 = x + (x^3)/3 + (x^5)/5 + ....

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    CA 16 07. Series of cosh(x)

    Use taylor theory
    • Since the derivative of cosh(x) is sinh(x) and derivative sinh(x) = cosh(x) hence
    • D0(0) = cosh(0) = 1
    • D1(0) = sinh(0) = 0
    • D2(0) = cosh(0) = 1
    • D3(0) = cosh(0) = 0
    • Hence cosh(x) = 1 + 0 + (x^2)/2! + 0 + (x^4)/4! +......
    • Hence sinh(x) = 1 + (x^2)/2! + (x^4)/4! + ....
    2nd method : cosh(x) = (e^(+x) + e(-x))/2
    • e^(+x) = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ......
    • e^(-x) = 1 - x + (x^2)/2! - (x^3)/3! + (x^4)/4! + ......
    • cosh(x) = ((e^(+x) + e^(-x))/2 = 1 + (x^2)/2! + (x^4)/4! + ....

    Go to Begin

    CA 18 08. Series of sin(x)
    Use Taylor theory
    • F(x) = sin(x) Hence F(0) = 0
    • D1(x) = +cos(x) Hence D1(0) = +1
    • D2(x) = -sin(x) Hence D2(0) = +0
    • D3(x) = -cos(x) Hence D3(0) = -1
    • D4(x) = +sin(x) Hence D4(0) = +0
    • D5(x) = +cos(x) Hence D5(0) = +1
    • D6(x) = -sin(x) Hence D6(0) = +0
    • D7(x) = -cos(x) Hence D7(0) = -1
    • D8(x) = +sin(x) Hence D8(0) = +0
    • sin(x) = F(0) + D1(0)*x + D2(0)*(x^2)/2! + D3(0)*(x^3)/3! + ......
    • sin(x) = x - (x^3)/3! + (x^5)/5! - .....
    Find sin(0.1 radians)
    • sin(0.1) = 0.1 - (0.1^3)/3! + (0.1^5)/5! - .....
    • sin(0.1) = 0.1 - 0.0001666 + 0.0000000
    • sin(0.1) = 0.099833
    • Note : This series converges very fast if x is small

    Go to Begin

    CA 18 09. Series of cos(x)

    Use Taylor theory
    • F(x) = cos(x) Hence F(0) = 1
    • D1(x) = -sin(x) Hence D1(0) = +0
    • D2(x) = -cos(x) Hence D2(0) = -1
    • D3(x) = +sin(x) Hence D3(0) = -0
    • D4(x) = +cos(x) Hence D4(0) = +1
    • D5(x) = -sin(x) Hence D5(0) = +0
    • D6(x) = -cos(x) Hence D6(0) = -1
    • D7(x) = +sin(x) Hence D7(0) = -0
    • D8(x) = +cos(x) Hence D8(0) = +1
    • sin(x) = F(0) + D1(0)*x + D2(0)*(x^2)/2! + D3(0)*(x^3)/3! + ......
    • sin(x) = 1 - (x^2)/2! + (x^4)/4! - .....
    Find cos(0.1 radians)
    • cos(0.1) = 1 - (0.1^2)/2! + (0.1^4)/4! - .....
    • cos(0.1) = 1 - 0.005 + 0.00000416
    • cos(0.1) = 0.9950042
    • Note : This series converges very fast if x is small

    Go to Begin

    CA 18 10. Series of arcsin

    Integral and arctan(x)
    • arcsin(x) = ∫(1/Sqr(1 - x^2))dx
    Expand 1/Sqr(1 - x^2) using binomial theory
    • 1/Sqr(1 - x^2)
    • = (1 - x^2)^(-1/2)
    • = 1 + (-1/2)*x^2 + (-1/2)*(-1/2-1)*((x^2)^2)/2! + (-1/2)*(-1/2-1)(-1/2-2)*((x^3)^3)/3! + ....
    • = 1 - (x^2)/2 + (-1)*(-3)*(x^4)/((2^2)*2!) + (-1)*(-3)(-5)*(x^6)/((2^3)*3!) + ...
    • = 1 - (x^2)/2 + 3*(x^4)/8 - 15*(x^6)/48 + ....
    Integral by power rule
    • arcsin(x) = ∫(1/Sqr(1 - x^2))dx
    • = ∫(1 - x^2 + x^4 - x^6 + x^8 - ....)dx
    • = x - (x^3)/6 + 3*(x^5)/40 - 15*(x^7)/336 + ....

    Go to Begin

    CA 18 00. Outlines

    Formula
    • Binomial theory
      • (x + y)^n = x^n + C(n,1)*(x^(n-1))*y + C(n,2)*(x^(n-2))*(y^2) + ....
      • C(n,r) = n*(n-1)*(n-2)*....*(n-r+1)/n!
    • Taylor theory
      • F(x) = F(0) + D1(0)*x + D2(0)*(x^2)/2! + ....
      • D1(0) is 1st derivative at x = 0
      • D2(0) is 2nd derivative at x = 0
      • D3(0) is 3rd derivative at x = 0
    Binomial theory method
    • 1. arctan(x) = x - (x^3)/3 + (x^5)/5 - ....
    • 2. ln(1 + x) = x - (x^2)/2 + (x^3)/3 - ....
    • 3. ln(1 - x) = x + (x^2)/2 + (x^3)/3 + ....
    • 4. arcsin(x) = x - (x^3)/6 + 3*(x^5)/40 - 15*(x^7)/336 + ....
    Taylor theory method
    • e^(+x) = 1 + x + (x^2)/2! + (x^3)/3! + ....
    • e^(-x) = 1 - x + (x^2)/2! - (x^3)/3! + ....
    • sin(x) = x - (x^3)/3! + (x^5)/5! - .....
    • cos(x) = 1 - (x^2)/2! + (x^4)/4! - .....
    • sinh(x) = x + (x^3)/3! + (x^5)/5! + ....
    • cosh(x) = 1 + (x^2)/2! + (x^4)/4! + ....
    Questions
    Go to Begin
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