
Power
series 
Maclaurin and Taylor series 
The
binomial series

The binomial
series expansions to the power series

The binomial
series expansion to the power series example






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
binomial series

We use the binomial theorem to expand
any positive integral power of a binomial (1
+ x)^{k},
as a polynomial with
k + 1
terms,


or
when writing the binomial coefficients in the shorter form



Assuming
f (x) =
(1
+ x)^{k},
where
k
is any real number, the function can be represented as a power
series using
the Maclaurin's formula,


Therefore,
we should calculate f (0)
and evaluate its successive derivatives at x =
0
to write f
^{(n) }(0),
so


Thus,
the Maclaurin's series for the power function


or 



is
called the binomial series.

We
can use the ratio test to examine its convergence, as


the binomial series
converges if  x
 < 1 and diverges if 
x  > 1.

Regarding
the endpoints, 1
and 1
of the interval of convergence, the series converges at 1
if 1
< k < 0
and
at both endpoints if k
> 0.


The binomial
series expansions to the power series

Hence,
for different values of k,
the binomial series


gives
the power series expansion of functions that we often use in
calculus, so

a)
for k
= 1


b)
for k
= 1/2


c)
for k
= 1/2


d)
for k
= 1/m



The binomial
series expansion to the power series example

Let's
graphically represent the power series of one of the above
functions inside its interval
of convergence. 

Example: Represent
f (x)
= 1/(1 + x^{2})
by the power series
inside the interval of convergence, graphically. 
Solution: As
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 then, 

let
write down the initial sequence of nth
order polynomials that describe the function inside the interval of convergence
1
< x < 1, 

Since
every polynomial above is missing the preceding odd degree term, their
coefficient a_{n}_{}_{1}
= 0 
thus, the coordinates
of translations 


Therefore,
the polynomials that describe the 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.


The above graph shows, all evenly indexed polynomials, with the positive
leading coefficient, intersect at (1,
1) and (1,
1) while all polynomials with odd
indexes, with the negative leading coefficient, intersect at (1,
0) and (1,
0). 
Thus, they will never reach the
functions values f
(1)
= 1/2 and f
(1) = 1/2, though their graphs
come closer and closer to the points (1,
1/2) and (1, 1/2) as n
increases. 










Calculus
contents B 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 