

Power
series

Power
series or polynomial with infinitely many terms 
The
sum of a power series is a function 
Maclaurin and Taylor series 
Representing
polynomial using Maclaurin's and Taylor's formula 
Representing
polynomial using Maclaurin's and Taylor's formula, examples 





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. 

Representing
polynomial using Maclaurin's and Taylor's formula 
If
given is an nth
degree polynomial 
P_{n
}(x) =
a_{n }x^{n} + a_{n}_{ }_{}_{
1}x^{n }^{}^{1} + ·
· · + a_{2
}x^{2}
+ a_{1 }x + a_{0 }then, 
P_{n}(0) =
a_{0},
P_{n}' (0) = a_{1},
P_{n}''(0) = 2!
a_{2},
P_{n}'''(0) = 3!
a_{3},
. . . , P_{n}^{(n)}(0) = n!
a_{n}
and
P_{n}^{(n
+ 1)} = 0 
so,
the coefficients of the polynomial 

therefore,
applying Maclaurin's formula, every polynomial can be written as 

since
P_{n}^{(n
+ 1)} = 0,
the remainder vanishes. 

Example: Represent
the quintic y
= 2x^{5} + 3x^{4} 
5x^{3} + 8x^{2} 
9x + 1 using Maclaurin's
formula. 
Solution: Let write
all successive derivatives of the given quintic function and evaluate them at the
origin, 
y' (x)
= 10x^{4} + 12x^{3} 
15x^{2} + 16x 
9,
y' (0)
= 
9 
y'' (x)
= 40x^{3} + 36x^{2} 
30x + 16,
y'' (0)
= 16 
y''' (x)
= 120x^{2} + 72x 
30,
y''' (0)
= 
30 
y^{IV}
(x)
= 240x + 72,
y^{IV}
(0)
= 72 
y^{V}
(x)
= 240,
y^{V}
(0)
= 240 
y^{VI}
= 0 and the last term of the polynomial a_{0}
= y(0)
= P_{5}(0) =
1, 
then
substitute obtained values into Maclaurin's
formula 


Example: Represent
the quartic y
= x^{4} 
4x^{3} + 4x^{2} + x 
4 at x_{0}
= 2 using
Taylor's
formula. 
Solution: Let write
all successive derivatives of the given quartic and evaluate them at x_{0}
= 2, 
y' (x)
= 4x^{3} 
12x^{2} + 8x + 1,
y' (2)
= 1 
y'' (x)
= 12x^{2} 
24x + 8,
y'' (2)
= 8 
y''' (x)
= 24x 
24,
y''' (2)
= 24 
y^{IV}
(x)
= 24,
y^{IV}
= 24 
y^{V}
(x)
= 0
and the last term, a_{0}
=
y(2)
= P_{4}(2) = 
2, 
then
substitute obtained values into Taylor's
formula 









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