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Power Series Expansion and Its Applications
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In the previous section, we discuss the convergence of power series, in its convergence region, the power series always converges to a function. For the simple power series, but also with itemized derivative, or quadrature methods, find this and function. This section will discuss another issue, for an arbitrary function , can be expanded in a power series, and launched into.
Whether the power series as and function? The following discussion will address this issue.3663
1 Maclaurin (Maclaurin) formula
Polynomial power series can be seen as an extension of reality, so consider the function can expand into power series, you can from the function and polynomials start to solve this problem. To this end, to give here without proof the following formula.
Taylor (Taylor) formula, if the function at in a neighborhood that until the derivative of order , then in the neighborhood of the following formula:
(9-5-1)
Among
That for the Lagrangian remainder. That (9-5-1)-type formula for the Taylor.
If so , get
, (9-5-2)
At this point,
( ).
That (9-5-2) type formula for the Maclaurin.
Formula shows that any function as long as until the derivative, can be equal to a polynomial and a remainder.
We call the following power series
(9-5-3)
For the Maclaurin series.
So, is it to for the Sum functions? If the order Maclaurin series (9-5-3) the first items and for , which
Then, the series (9-5-3) converges to the function the conditions
.
Noting Maclaurin formula (9-5-2) and the Maclaurin series (9-5-3) the relationship between the known
Thus, when
There,
Vice versa. That if
,
Units must
.
This indicates that the Maclaurin series (9-5-3) to and function as the Maclaurin formula (9-5-2) of the remainder term (when ).
In this way, we get a function the power series expansion:
. (9-5-4)
It is the function the power series expression, if, the function of the power series expansion is unique. In fact, assuming the function f(x) can be expressed as power series
, (9-5-5)
Well, according to the convergence of power series can be itemized within the nature of derivation, and then make (power series apparently converges in the point), it is easy to get
.
Substituting them into (9-5-5) type, income and the Maclaurin expansion of (9-5-4) identical.
In summary, if the function f(x) contains zero in a range of arbitrary order derivative, and in this range of Maclaurin formula in the remainder to zero as the limit (when n → ∞,), then , the function f(x) can start forming as (9-5-4) type of power series.
Power Series
,
Known as the Taylor series.
Second, primary function of power series expansion
Maclaurin formula using the function expanded in power series method, called the direct expansion method.
Example 1
Test the function expanded in power series of .
Solution because
,
Therefore
,
So we get the power series
, (9-5-6)
Obviously, (9-5-6)type convergence interval , As (9-5-6)whether type is Sum function, that is, whether it converges to , but also examine remainder .