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Mathematics Dictionary
Dr. K. G. Shih
Exponent Series of e^x


  • Q01 | - Series of y = e^x
  • Q02 | - Find series of sin(x)
  • Q03 | - Find series of cos(x)
  • Q04 | - Find series of sinh(x)
  • Q05 | - Find series of cosh(x)

    • Q01. Series of e^x

    Taylor's expansion
    • F(x) = F(0) + F1(0)*x + F2(0)*(x^2)/(2!) + F3(0)*(x^3)/(3!) + ....
    • Where
      • F1(0) = 1 is 1st derivative of e^x for x = 0
      • F2(0) = 1 is 2nd derivative of e^x for x = 0
      • F3(0) = 1 is 3rd derivative of e^x for x = 0
      • F4(0) = 1 is 4th derivative of e^x for x = 0
      • Etc.
    • Hence e^x = 1 + x + (x^2)/(2!) + (x^3)/(3!) + .....
    Value of e
    • e = 1 + 1 + 1/(2!) + 1/(3!) + .... = 2.718281829....

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    Q02. Series of sin(x)

    Method 1 : Using e^(i*x) = cos(x) + i*sin(x)
    • Value of i
      • i^2 = -1
      • i^3 = -i
      • i^4 = +1
      • i^5 = +i
    • e^(i*x)
    • = 1 + i*x + ((i*x)^2)/(2!) + ((i*x)^3)/(3!) + ....
    • = (1 - (x^2)/(2!) + (x^4)/(4!) - ...) + i*(x - (x^3)/(3!) + (x^5)/(5!) - ...)
    • = cos(x) + i*sin(x)
    • Hence sin(x) = (x - (x^3)/(3!) + (x^5)/(5!) - (x^7)/(7!) + ......)
    Method 2 : Taylor's expansion
    • F(x) = F(0) + F1(0)*x + F2(0)*(x^2)/(2!) + F3(0)*(x^3)/(3!) + ....
    • Where
      • F1(0) = +cos(0) = +1 is 1st derivative of sin(x) for x = 0
      • F2(0) = -sin(0) = -0 is 2nd derivative of sin(x) for x = 0
      • F3(0) = -cos(0) = -1 is 3rd derivative of sin(x) for x = 0
      • F4(0) = +sin(0) = +0 is 4th derivative of sin(x) for x = 0
      • F5(0) = +cos(0) = +1 is 5th derivative of sin(x) for x = 0
      • Etc.
    • Hence sin(x) = x - (x^3)/(3!) + (x^5)/(5!) - ......
    Note
    • 1st derivative of sin(x) = +cos(x)
    • 2nd derivative of sin(x) = -sin(x)
    • 3rd derivative of sin(x) = -cos(x)
    • 4th derivative of sin(x) = +sin(x)
    • 5th derivative of sin(x) = +cos(x)
    • 6th derivative of sin(x) = -sin(x)
    • 7th derivative of sin(x) = -cos(x)
    • 8th derivative of sin(x) = +sin(x)

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    Q03. Series of cos(x)

    Method 1 : Using e^(i*x) = cos(x) + i*sin(x)
    • Value of i
      • i^2 = -1
      • i^3 = -i
      • i^4 = +1
      • i^5 = +i
    • e^(i*x)
    • = 1 + i*x + ((i*x)^2)/(2!) + ((i*x)^3)/(3!) + ....
    • = (1 - (x^2)/(2!) + (x^4)/(4!) - ...) + i*(x - (x^3)/(3!) + (x^5)/(5!) - ...)
    • = cos(x) + i*sin(x)
    • Hence cos(x) = (1 - (x^2)/(2!) + (x^4)/(4!) - (x^6)/(6!) + ......)
    Method 2 : Taylor's expansion
    • F(x) = F(0) + F1(0)*x + F2(0)*(x^2)/(2!) + F3(0)*(x^3)/(3!) + ....
    • Where
      • F1(0) = -sin(0) = +0 is 1st derivative of cos(x) for x = 0
      • F2(0) = -cos(0) = -1 is 2nd derivative of cos(x) for x = 0
      • F3(0) = +sin(0) = -0 is 3rd derivative of cos(x) for x = 0
      • F4(0) = +cos(0) = +1 is 4th derivative of cos(x) for x = 0
      • F5(0) = -sin(0) = -0 is 5th derivative of cos(x) for x = 0
      • Etc.
    • Hence cos = 1 - (x^2)/(2!) + (x^4)/(4!) - ......
    Note
    • 1st derivative of sin(x) = +cos(x)
    • 2nd derivative of sin(x) = -sin(x)
    • 3rd derivative of sin(x) = -cos(x)
    • 4th derivative of sin(x) = +sin(x)
    • 5th derivative of sin(x) = +cos(x)
    • 6th derivative of sin(x) = -sin(x)
    • 7th derivative of sin(x) = -cos(x)
    • 8th derivative of sin(x) = +sin(x)
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    Q04. Series of sinh(x)

    Method 1 : Using sinh(x) = ((e^x) - e^(-x))/2
    • sinh(x)
    • = ((1 + x + (x^2)/(2!) + (x^3)/(3!) + ...) - (1 - x + (x^2)/(2!) - ...)/2
    • = x + (x^3)/(3!) + (x^5)/(5!) + .....
    Method 2 : Taylor's expansion
    • F(x) = F(0) + F1(0)*x + F2(0)*(x^2)/(2!) + F3(0)*(x^3)/(3!) + ....
    • Where
      • F1(0) = +cosh(0) = +1 is 1st derivative of sinh(x) for x = 0
      • F2(0) = +sinh(0) = -0 is 2nd derivative of sinh(x) for x = 0
      • F3(0) = +cosh(0) = +1 is 3rd derivative of sinh(x) for x = 0
      • F4(0) = +sinh(0) = +0 is 4th derivative of sinh(x) for x = 0
      • F5(0) = +cosh(0) = +1 is 5th derivative of sinh(x) for x = 0
      • Etc.
    • Hence sinh(x) = x + (x^3)/(3!) + (x^5)/(5!) + ......
    Note
    • 1st derivative of sinh(x) = +cosh(x)
    • 2nd derivative of sinh(x) = +sinh(x)
    • 3rd derivative of sinh(x) = +cosh(x)
    • 4th derivative of sinh(x) = +sinh(x)
    • 5th derivative of sinh(x) = +cosh(x)
    • 6th derivative of sinh(x) = +sinh(x)
    • 7th derivative of sinh(x) = +cosh(x)
    • 8th derivative of sinh(x) = +sinh(x)

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    Q05. Series of cosh(x)

    Method 2 : Using cosh(x) = ((e^x) + e^(-x))/2
    • cosh(x)
    • = ((1 + x + (x^2)/(2!) + (x^3)/(3!) + ...) + (1 - x + (x^2)/(2!) - ...)/2
    • = 1 + (x^2)/(2!) + (x^4)/(4!) + .....
    Method 2 : Taylor's expansion
    • F(x) = F(0) + F1(0)*x + F2(0)*(x^2)/(2!) + F3(0)*(x^3)/(3!) + ....
    • Where
      • F1(0) = +sinh(0) = +1 is 1st derivative of cosh(x) for x = 0
      • F2(0) = +cosh(0) = +0 is 2nd derivative of cosh(x) for x = 0
      • F3(0) = +sinh(0) = +1 is 3rd derivative of cosh(x) for x = 0
      • F4(0) = +cosh(0) = +0 is 4th derivative of cosh(x) for x = 0
      • F5(0) = +sinh(0) = +1 is 5th derivative of cosh(x) for x = 0
      • Etc.
    • Hence sinh(x) = x + (x^3)/(3!) + (x^5)/(5!) + ......
    Note
    • 1st derivative of cosh(x) = +sinh(x)
    • 2nd derivative of cosh(x) = +cosh(x)
    • 3rd derivative of cosh(x) = +sinh(x)
    • 4th derivative of cosh(x) = +cosh(x)
    • 5th derivative of cosh(x) = +sinh(x)
    • 6th derivative of cosh(x) = +cosh(x)
    • 7th derivative of cosh(x) = +sinh(x)
    • 8th derivative of cosh(x) = +cosh(x)

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