
Integral
calculus 


Differentiation
and integration of infinite series

Differentiation
and integration of infinite series examples

Differentiation
of power series 





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
and integration of infinite series examples

Example:
Represent the f
(x) = arctan x
or f
(x) = tan ^{}^{1}
x,
by a power series.

Solution:
Since 


thus, we should integrate the series


Let
write down the initial sequence of nth
order polynomials that describe the function inside the interval of convergence
1
< x < 1, 

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.


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. 
Then by integrating the series


and since for
x = 0
the integral is zero then, C
= 0 therefore


From where, for x
= 1


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, 

Since
every polynomial above is missing the preceding even 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 odd degree, as
shows the picture below.


Note that all polynomials in the
series with odd indexes have extreme points at x
= 1
and x =
1.


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).









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