Power series
      Maclaurin and Taylor series
      The power series expansion of the sine function
         Properties of the power series expansion of the sine function
      The power series expansion of the cosine function
         Properties of the power series expansion of the cosine function
Maclaurin and Taylor series
Consider the polynomial function
f (x) = an xn + an - 1xn - 1 + a3 x3 + a2 x2 + a1 x + a0.
If we write the value of the function and the values of its successive derivatives, at the origin, then
f (0) = a0,     f '(0) = 1 a1,     f ''(0) = 1 2a2,     f '''(0) = 1 2 3a3,  . . .  ,  f (n) (0) = n! an  
so we get the coefficients;
Then, the polynomial f (x) with infinitely many terms, written as the power series
and
where 0! = 1,   f (0)(x0) =  f(x0) and  f (n)(x0) is the nth derivative of  f at x0,
represents an infinitely differentiable function and is called Maclaurin series and Taylor series respectively.
The power series expansion of the sine function
Example:  Let represent the sine function  f(x) = sin x by the infinite polynomial (or power series).
Solution:  The sine function is the infinitely differentiable function defined for all real numbers. 
We use the polynomial with infinitely many terms
to represent the sine function. We should calculate the function value  f (0), and some successive derivatives of the sine function, to determine the nth order derivative, therefore
Obtained values  f (0)  and  f (n) (0) substituted into Maclaurin's formula yield,
Properties of the power series expansion of the sine function
The polynomials that describe the sine function all are source polynomials of odd degree. Meaning, both
coordinates of translations  are zero since all even derivatives  are
zero, the infinite polynomial is missing every next even term, therefore  an-1 = 0.
Notice that every second polynomial in the above sequence, whose leading term is negative, represents the variant  f (-x) (or  - f (x)) of the source polynomial of odd degree whose x or y variable changed the sign, as are the graphs,  f3 (x) and  f7 (x).
The power series expansion of the cosine function
Example:  Let represent the cosine function  f (x) = cos x by the infinite polynomial (or power series).
Solution:  The cosine function is the infinitely differentiable function defined for all real numbers. 
We should calculate the function value  f (0), and some successive derivatives of the cosine function, to determine the nth order derivative, therefore
Obtained values  f (0) and  f (n) (0) substituted into Maclaurin's formula yield,
Properties of the power series expansion of the cosine function
Since all odd derivatives are zero, the infinite polynomial is missing every next odd term, therefore  an-1 = 0
and the coordinates of translations  
The polynomials that describe the cosine function all are source polynomials of even degree translated in the direction of the y axis by  y0 = 1, as is shown in the picture below.
Notice that every second polynomial in the above sequence, whose leading term is negative, represents the variant  - f (x) of the source polynomial of even degree, as are the graphs,  f2 (x) and  f6 (x).
Calculus contents B
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