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Taylor series was one of the importance achievement of human science.
We tried to get the approximation for an arbitray function by
polynomials. Since, with polynomials, we know how todifferentiate,
integrate, find the roots,draw the graphs,etc.
The results of Taylor's series is,
for a nth differentiable function f(x)
f(x) = E f^(k)(a)(x-a)^k/k! ,
(the series converges to f(x) locally at a)
when a = 0, we have
f(x) = E f^(k)(0)x^k/k!
Without this series, we cannot use any calculators or computers
to find the value of sine,cosine,log or expontial, ect.
In other words, there is no modern science without Taylor series.
Now, f(x) = sin x, expand it at x = 0,
we have f'(x) = cos x, f'(x) = -sin x, f"'(x) = -cos x,
so f'(0) = 1, f"(0) = 0, f"'(0) = -1,
after simplified, we have
sin x = E (-1)^(2k+1))x^(2k+1)/(2k+1)!
or sin x = x -x^3/3! + x^5 /5! -x^7/7! +....
I used the first two terms to estimate sinx before.
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