Integral calculus
      Differentiation and integration of infinite series
         Differentiation of power series
         Differentiation and integration of infinite series example
Differentiation and integration of infinite series
If  f (x) is represented by the sum of a power series
with radius of convergence r > 0 and  - r < x < r, then the function has the derivative
and the function has the integral
Thus, a power series can be differentiated and integrated term by term while the radius of convergence remains the same, with only (possible) exception at the endpoints of the interval of convergence.
Differentiation of power series
Recall that the exponential function  f (x) = ex represented by the power series
is absolutely convergent for all real x since
by the ratio test
the limit L < 1 for any value of x.
Applying the power rule
thus, for all real x the function f (x) = ex is equal to its own derivative  f ' (x).
Example:   Find   by representing the integrand function as the power series.
Solution:  By substituting  - x2 for x in the above power series expansion of  ex we get
Let write down the initial sequence of nth order polynomials that describe the function for all real x,
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.
Note that the roots of odd indexed polynomials in the series correspond to the abscissas of successive even indexed polynomials, as shows the above graph. 
On the graph of the bell-shaped curve, representing the probability density function of a normal distribution, at x = 2/2 denoted are the points of inflections.
Therefore, the power series representing the normal curve converges for all real x.
Hence, by integrating the series term by term obtained is
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