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Integral
calculus |
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Differentiation
and integration of infinite series
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Differentiation
of power series |
Differentiation
and integration of infinite series example
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Differentiation
and integration of infinite series
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If
f (x)
is represented by the sum of a power series
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with
radius of convergence r
> 0 and -
r < x < r,
then the function has the derivative
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and
the function has the integral
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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.
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Differentiation
of power series |
Recall
that the exponential function f
(x)
= ex
represented
by the power
series
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is absolutely convergent for all real x
since |
by
the ratio test |
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the
limit L
< 1 for
any value of x. |
Applying the power rule
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thus, for all real x
the function f (x)
= ex is equal
to its own derivative f
' (x).
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Example:
Find |
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by
representing
the integrand function as the power series. |
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Solution: By
substituting -
x2
for x
in the above power series expansion of ex
we get |
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Let
write down the initial sequence of nth
order polynomials that describe the function for all real x, |
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Since
every polynomial above is missing the preceding odd degree term, their
coefficient an-1
= 0 |
thus, the coordinates
of translations |
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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.
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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.
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On the graph of the bell-shaped curve,
representing the probability density function of a normal
distribution, at
x = ±
Ö2/2
denoted are the points of
inflections.
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Therefore, the power series representing the
normal curve converges for all real x.
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Hence, by integrating the series term
by term obtained is
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Functions
contents G |
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