
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) = a_{n
}x^{n}
+ a_{n}_{ }_{}_{
1}x^{n }^{}^{
}^{1} + · · · + a_{3 }x^{3}
+ a_{2 }x^{2} + a_{1 }x
+ a_{0}.

If
we write the value of the function and the values of its successive
derivatives, at the origin, then 
f
(0) =
a_{0}, f '(0) = 1· a_{1},
f ''(0) = 1· 2a_{2},
f '''(0) = 1· 2· 3a_{3}, . . . , f ^{(}^{n}^{)
}(0) =
n! a_{n} 
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)}(x_{0})
= f(x_{0})
and
f ^{(}^{n}^{)}(x_{0})
is the nth
derivative of f at
x_{0}, 
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 a_{n}_{}_{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, f_{3
}(x)
and f_{7 }(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 a_{n}_{}_{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 y_{0}
= 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, f_{2
}(x)
and f_{6 }(x). 









Calculus
contents B 



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