2.8. Usage of Taylor’s series
Calculation of trigonometric function’s values. To find some values of sine and cosine functions we use corresponding Taylor’s series.
Example 2.11. Compute sin1:
Here
.
2. Root’s calculation. To find some root’s values we use binomial series:
Example
2.12. To
find
we transform it in the following way:
And
obtain
.
Thus
,
Here
.
After computing we have
.
3.
Evaluation of natural logarithms. To
obtain Maclaurin series for
we consider its derivative
as the sum of geometrical progression with the ratio
After
integration we have Maclaurin series for
If we substitute –x after x, we obtain another series:
Subtracting these two series we obtain the series more convenient for evaluation of natural logarithms:
Example
2.13. Compute
.
Solution.
To
find
,
we substitute
after
,
so we have the number series :
2.9. Evaluating “inexpressible” integrals
One
of the antiderivatives
of the elementary function
can be written with help of the definite integral with a variable
upper limit
,
which belongs to the class of “inexpressible” integrals.
The
thing is, that the antiderivatives of the function
are not elementary functions.
But,
can be represented in the form of finite power series.
Substituting
for the
into the Taylor’s series of the function
we find:
.
Since a power series can be integrated termwise, we have
.
Example
2.14. Find
solution
to evaluate the integral
,
We use the series:
If we confine ourselves to the first three terms of the series and evaluate, then we get:
So
that
We
note, that in evaluating the given integral by rectangle or trapezoid
method, such an accuracy is attained only with a number of terms
.
2.10. Integration of differential equations
Consider
the Cauchy problem
Suppose
that the function
is analytic in the neighborhood of the point
i.e., is represent able a power series in
,
and
.
The solution
)
of the Cauchy problem can then be obtained as the series
Knowing
we will find
;
Differentiating the equation with respect to x gives:
.
Substituting
into the right-hand side the value
and the value
we have just found, we will have
,
and so on.
Example 2.15. Find the solution of the Cauchy problem
We
have
.
Differentiation
gives
If we apply differentiation several times we will obtain
And
so on. Thus we have
2.11. Power series with complex terms
If it is possible to consider power series of the form
Where
are given complex numbers, and the variable
takes on complex values.
Series
is said to converge at the point
,
if there is a complex number
(called the sum of the series), such that for any real
there is a number
such that for any
inequality
Will be satisfied. The set of all for which series converges is called the domain of convergence.
Power
series converges either at the point
,
or at any point
of the complex plane, or there exists a number
such that in a circle
the series converges, and at any point
satisfying the inequality
<
diverges (compare with 2.4.).
Example
2.16. For
the series
,
the radius
,
we have convergence at the point
,
(here
).
Example
2.17. The
series
Is
convergent at any point of the complex plane, because the radius of
convergence
