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Australopithecus

  • 2 years ago

Use the Maclaurin Series for f(x) using the definition of the Maclaurin Series for sin(pix) Can anyone show me how to find the series using the Maclaurin method?

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  1. 91
    • 2 years ago
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    f(0) f'(0)(x-0) f''(0)(x-0)^2 ------- + ---------- + ------------- 0! 1! 2!

  2. Australopithecus
    • 2 years ago
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    I just memorized the table the answer is, but it would be nice to know the method to this as it will probably come up on my final \[\sum_{n=0}^{\infty} \frac{(-1)^{n}(\pi x)^{2n+1}}{(2n+1)!}\]

  3. 91
    • 2 years ago
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    so let's start taking derivative

  4. 91
    • 2 years ago
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    f'(0)= pi cos(pi x) = pi f''(0)=-pi^2 sin(pi x)=0 f'''(0)=-pi^3 cos(pi x)=-pi^3

  5. 91
    • 2 years ago
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    so you see the pattern pi, 0, -pi^3,0,pi^5,0,-pi^7

  6. Australopithecus
    • 2 years ago
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    right

  7. 91
    • 2 years ago
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    so you can put it into the series format pi x - pi^3 (x)^3 + pi^5 (x)^5 --------- ----------- - ..... 3! 5!

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