
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)
= e^{x}
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)
= e^{x} is equal
to its own derivative f
' (x).


Example:
Find 

by
representing
the integrand function as the power series. 

Solution: By
substituting 
x^{2}
for x
in the above power series expansion of e^{x}
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 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.


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 bellshaped curve,
representing the probability density function of a normal
distribution, at
x = ±
Ö2/2
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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