Power series
      Maclaurin and Taylor series
      The power series expansion of the exponential function
         Properties of the power series expansion of the exponential function
         The radius of convergence or the interval of convergence
Maclaurin and Taylor series
Consider the polynomial function
f (x) = an xn + an - 1xn - 1 + a3x3 + a2x2 + a1x + 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 power series expansion of the exponential function
Example:  Let represent the exponential function  f (x) = ex 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  ex so that,   f (0) = e0 = 1,  f (n)(0) = e0 = 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,  f1(x)f2(x)f3(x), . . . fn(x), represents translation of its original or source function that passes through the origin, we calculate the coordinates of translations, x0 and  y0,  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 x0.
Below listed sequence of the polynomials and corresponding vertical translations y0 shows that their graphs approach closer and closer to the point (x0, y0) 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 + y0 = an(x + x0)n + an - 1(x + x0)n - 1 + + a2(x + x0)2 + a1(x + x0) + a0,
after expanding and reducing the expression, we get the source polynomial function passing through the origin. The above expression we can write as
fs(n)(x) + y0fn(x + x0)     or     fs(n)(x) =  fn(x + x0) - fn(x0),
For example, we obtain the source quadratic fs2(x) from the expression
fs2(x) + y0f2(x + x0)     or     fs2(x) =  f2(x + x0) - y0  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 x0 and y0 into the source polynomial to get its translated form or, we can check if the derivative at x0 = - 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 f1 to  fn in the above sequence have the same horizontal translation x0 = - 1.
Recall that an nth degree polynomial function and all its successive derivatives to the (n - 1)th order, have constant horizontal translation x0.
Since every polynomial in the above sequence represents the derivative of its successor, that is, 
f 'n (x)fn - 1(x)   and thus   f 'n (x0)fn - 1(x0).
Therefore as consequence, each x-intercept 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, a1, a2, and a3 of the source polynomial  fs5(x) are
Hence, the vertical translations y0 of the successive derivatives are,
  f4(x0) = 1! f5(x0) = 3/8,    f3(x0) = 2! f5(x0) = 1/3   and   f2(x0) = 3! f5(x0) = 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  
The radius of convergence or the interval of convergence
  - If the power series  anxn  converges when x = x1, then it converges for every x that is closer to the
    origin than x1, that is, whenever  | x | < | x1 |.
  - If the power series  anxn  diverges when x = x1, then it diverges for every x that is further from the origin
    than x1, that is, whenever  | x | > | x1 |.
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 | < 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  anxn  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  an(x - x0)n  converges absolutely at every point x from the open interval |x - x0| < r or
x0 - r < x < x0 + r   and diverges for all x outside this interval.
Note that the series may or may not converge when | x - x0 | = r that is, when  x = x0 - r  or  x = x0 + 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.
Calculus contents B
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