
Series 
Power
series

Maclaurin and Taylor
series 
The power series expansion of the exponential
function 
Properties
of the power series expansion of the exponential function 







Power
series or polynomial with infinitely many terms 
A
real power series in x
around the origin (or centered at the origin) is a series of functions of the form 

and
the power series around a given point x
= x_{0} (or centered
at x_{0})
is a series of the form 

where
the coefficients a_{n}
are fixed real numbers and x
is a real variable. 
A
power series with real coefficients is said to be real or complex
according as both x
and x_{0}
are real or complex
numbers. 
Therefore,
the nth
partial sum of a power series is a polynomial of degree n, 


The
sum of a power series is a function 
The
sum of a power series is a function 

the domain of
which
is the set of those values of x
for which the series converges to the value of the 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 exponential function 
Example: Let
represent the exponential function f
(x)
= e^{x}
by the infinite polynomial (power series). 
Solution: The
exponential function is the infinitely differentiable function defined
for all real numbers whose 
derivatives
of all orders are equal e^{x}
so that, f (0)
= e^{0} = 1,
f ^{(}^{n}^{)}(0)
= e^{0} = 1 and 

therefore 

the function can be represented as a power
series using
the Maclaurin's formula 

the exponential function
is represented
by the power series that is absolutely convergent for all real x
since 
by
the ratio test 


the
limit L
< 1 for
any value of x. 


Properties
of the power series expansion of the exponential function 
Since
every polynomial function in the sequence, f_{1
}(x),
f_{2 }(x),
f_{3 }(x),
. . . , f_{n
}(x),
represents translation of its
original or source function that passes through the origin, we calculate the coordinates of
translations, x_{0} and
y_{0},
to get their source forms. 
Now
we apply the method and formulas that are revealed and
explored under the 'Polynomial' sections.
Thus, the coordinates of translations, 

Note
that the above result proves the main property of the polynomial stating
that, an nth
degree polynomial function and all its successive derivatives to the (n

1)th order, have constant horizontal translation
x_{0}. 
Below
listed sequence of the polynomials and corresponding vertical
translations y_{0}
shows that their graphs
approach closer and closer to the point (x_{0},
y_{0}) or (1,
1/e) as n
tends to infinity. 

By
plugging the coordinates of translations with changed signs into the
polynomial expressed in the general 
form
y + y_{0}
= a_{n }(x + x_{0})^{n}
+ a_{n }_{}_{
1}(x + x_{0})^{n }^{}^{
1} + · · · + a_{2 }(x +
x_{0})^{2} + a_{1 }(x + x_{0})
+ a_{0}, 
after
expanding and reducing the expression, we get the source polynomial
function passing through the origin.
The above expression we can write as 
f_{s(n)
}(x)
+ y_{0} = f_{n }(x + x_{0})
or f_{s(n)
}(x)
= f_{n }(x + x_{0}) 
f_{n }(x_{0}), 
For
example, we obtain the source quadratic f_{s2
}(x)
from the expression 
f_{s2
}(x)
+ y_{0} = f_{2 }(x + x_{0})
or f_{s2
}(x)
= f_{2 }(x + x_{0}) 
y_{0} that is, 

The
same way we get the source function of every polynomial as listed below. 

To
prove that expressions on the left and the right side of the same row
represent the same curve plug x_{0} and
y_{0}
into the source polynomial to get its translated form or, we can check
if the derivative
at x_{0}
=  1
of the
left side polynomial
is
equal to the derivative at x
= 0 of the source polynomial
(the graph of which passes
through the origin) on the right side. Thus,
for example 

Note
that all polynomials from f_{1}
to f_{n}
in the above sequence have the same horizontal translation x_{0}
=  1.

Recall
that an nth
degree polynomial function and all its successive derivatives to the (n

1)th order, have constant
horizontal translation x_{0}. 
Since
every polynomial in the above sequence represents the derivative of its
successor, that is, 
f
'_{n}(x)
= f_{n }_{}_{
1}(x) and thus
f
'_{n}(x_{0})
= f_{n }_{}_{
1}(x_{0}). 
Therefore
as consequence, each xintercept
of odd polynomial in the sequence determines the abscissa of the
only extreme (minimum) of succeeding even polynomial and the abscissa of
the only point of inflection of succeeding
odd polynomial, as shows the picture above. 
In
the same way, for example the coefficients, a_{1},
a_{2},
and a_{3}
of the source polynomial f_{s5}(x)
are 

Hence,
the vertical translations y_{0}
of the successive derivatives are, 
f_{4}(x_{0})
= 1! f_{5}´(x_{0})
= 3/8, f_{3}(x_{0})
= 2! f_{5}´´(x_{0})
= 1/3 and
f_{2}(x_{0})
= 3! f_{5}´´´(x_{0})
= 1/2 , 
as
is shown above. 
Thus,
the power series expansion of the exponential function for x
= 1 yields 


from
where we get the number e
as accurate as we please. 

For
x = 1
it yields 

and similarly 












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