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| Integral
calculus |
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Differentiation
and integration of infinite series
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Differentiation
and integration of infinite series examples
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Differentiation
of power series |
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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
and integration of infinite series examples
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Example:
Represent the f
(x) = arctan x
or f
(x) = tan -1
x,
by a power series.
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| Solution:
Since |
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thus, we should integrate the series
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| Let
write down the initial sequence of nth
order polynomials that describe the function inside the interval of convergence
-1
< x < 1, |
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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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| The above graph
shows that all evenly indexed polynomials (with the positive
leading coefficient) intersect at (-1,
1) and (1,
1) while all polynomials with odd
indexes, with the negative leading coefficient, intersect at
(-1,
0) and (1,
0). |
| Thus, they will never reach the
functions values f
(-1)
= 1/2 and f
(1) = 1/2, though their graphs
come closer and closer to the points (-1,
1/2) and (1, 1/2) as n
increases. |
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Then by integrating the series
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and since for
x = 0
the integral is zero then, C
= 0 therefore
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From where, for x
= 1
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| Let
write down the initial sequence of nth
order polynomials, which describe the f
(x) = arctan x
by the above
power series inside the interval of convergence
-1
< x < 1, |
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| Since
every polynomial above is missing the preceding even 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 odd degree, as
shows the picture below.
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Note that all polynomials in the
series with odd indexes have extreme points at x
= -1
and x =
1.
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| Differentiation
of power series |
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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 |
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the ratio test |
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limit L
< 1 for
any value of x. |
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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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| Functions
contents G |
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