

Power
series

Power
series or polynomial with infinitely many terms 
The
sum of a power series is a function 
Maclaurin and Taylor series 
The radius of convergence or the interval of convergence 





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 radius of convergence or the interval of
convergence 
 If the power series a_{n}x^{n}
converges when x = x_{1},
then it converges for every x
that is closer to the 
origin than x_{1},
that is, whenever  x
 <  x_{1} .

 If the power series a_{n}x^{n}
diverges when x = x_{1},
then it diverges for every x
that is further from the origin 
than x_{1},
that is, whenever  x
 >  x_{1} . 
The
real power series converges for all absolute value of x
that are less than a number r,
called the radius of convergence or the
interval of convergence,
written  x 
< r that is, the open interval

r < x < r. 
We
apply the root test or the ratio test to find the interval of
convergence. Thus, 


the power series 

converges, 




the power series diverges. 


Therefore
the inequality, 


defines the interval
in which the power series is absolutely convergent. 
Thus,
denoting the right side of the above inequality by r,
we get
the interval of convergence
 x  < r
saying, for
every x
between 
r
and r
the series
converges absolutely while, for
every x
outside that interval the series
diverges.

Therefore,
the radius of convergence of the power series, 

The power series
a_{n}x^{n}
converges absolutely at every point x
from the open interval 
r < x < r and
diverges for all
x outside this interval. At
the endpoints 
r
and r,
the series may converge or diverge so these
points must be checked for convergence, individually. 
The power series
a_{n}(x

x_{0})^{n}
converges absolutely at every point x
from the open interval x

x_{0}
< r or 
x_{0}

r < x < x_{0}
+ r and
diverges for all
x outside this interval.

Note
that the series may or may not converge when  x

x_{0 } = r
that is, when x
=
x_{0}

r
or x
=
x_{0}
+
r.

To
determine whether the power series converge or diverge at endpoints we
should plug each endpoint into the
given series and apply appropriate test for convergence. 








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