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) = an xn + an - 1 xn - - 1 + a3 x3 + a2 x2 + a1 x + a0.
If we write the value of the function and the values of its successive derivatives, at the origin, then
f (0) = a0,     f '(0) = 1 a1,     f ''(0) = 1 2a2,     f '''(0) = 1 2 3a3,  . . .  ,  f (n) (0) = n! an  
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) (x0) =  f (x0) and  f (n) (x0 is the nth derivative of  f at x0,
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 + x2) 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  an-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  y0 = 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
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