Power series
      Maclaurin and Taylor series
      The power series expansion of the logarithmic function
         Properties of the power series expansion of the logarithmic function
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 (x0and  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 logarithmic function
Example:  Let represent the translated (shifted) logarithmic function  f (x) = ln (x + 1) by the power series.
Solution:  Given translated logarithmic function is the infinitely differentiable function defined for all
 - 1 < x < oo . We use the polynomial with infinitely many terms in the form of power series
to represent given function. We should calculate the function value  f (0), and some successive derivatives of
the logarithmic function, to determine the nth order derivative.
Properties of the power series expansion of the logarithmic function
To graphically represent the power series (or polynomial) expansion of the logarithmic function we first calculate the coordinates of translations of the given sequence of polynomials. Since,
The sequence of the polynomials and corresponding translations are written below
and in the picture below shown are their graphs, where are also marked the (x0, y0) points that approach the  origin as n tends to infinity.
Inside the interval  -1 < x < 1  the graphs of the polynomials approach closer and closer to the logarithmic function, as n increases, but for x > 1 the graphs of even and odd degree polynomials separate as shows the picture above.
By plugging the coordinates of translations with changed signs into the polynomial expressed in the general form, we get the source polynomial function. Thus, the source functions of the above polynomials are
We can apply the ratio test to check the interval of convergence.
Since the power series converges at every point x from the open interval  | x | < r  or  - r < x < r  and diverges for all x outside this interval, where
therefore | x | < r,   | x | < 1  or the power series converges in the open interval   - 1 < x < 1, and diverges outside this interval.
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.
The f (x) = ln (x + 1) is not defined at x = -1, so we only test for convergence the right side endpoint by plugging it into given power series.
Thus,  
Calculus contents B
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