| 单词 | power series |
| 例句 | In addition, power series can be easily differentiated and integrated, thus being useful in solving differential equations and integrating complicated functions. Calculus, Vol 2 2016-03-30T00:00:00Z A polar point of indetermination is a point about which the function can be expressed as a quotient of two converging power series, both of which vanish at the point. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z A method was given by J. H. Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series. Encyclopaedia Britannica, 11th Edition, Volume 7, Slice 2 "Constantine Pavlovich" to "Convention" However, we can find a power series for f . Calculus, Vol 2 2016-03-30T00:00:00Z We show how power series can be used to evaluate integrals involving functions whose antiderivatives cannot be expressed using elementary functions. Calculus, Vol 2 2016-03-30T00:00:00Z As we will see in this chapter, representing functions using power series allows us to solve mathematical problems that cannot be solved with other techniques. Calculus, Vol 2 2016-03-30T00:00:00Z As regards power series we have the following theorems: 1. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z One way to evaluate such integrals is by expressing the integrand as a power series and integrating term by term. Calculus, Vol 2 2016-03-30T00:00:00Z In this section we define power series and show how to determine when a power series converges and when it diverges. Calculus, Vol 2 2016-03-30T00:00:00Z We also show how to represent certain functions using power series. Calculus, Vol 2 2016-03-30T00:00:00Z A power series is a type of series with terms involving a variable. Calculus, Vol 2 2016-03-30T00:00:00Z The theory of power series has been developed chiefly from the point of view of the theory of functions of complex variables. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z For a power series centered at x = a, the value of the series at x = a is given by c0. Calculus, Vol 2 2016-03-30T00:00:00Z Some power series converge only at that value of x. Calculus, Vol 2 2016-03-30T00:00:00Z Most power series, however, converge for more than one value of x. Calculus, Vol 2 2016-03-30T00:00:00Z As a result, a power series can be thought of as an infinite polynomial. Calculus, Vol 2 2016-03-30T00:00:00Z This function is analytic in the domain, and the power series representing it is the Taylor’s series for the function. Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 3 "Frost" to "Fyzabad" 2011-08-15T02:00:28.473Z We now summarize these three possibilities for a general power series. Calculus, Vol 2 2016-03-30T00:00:00Z To determine the interval of convergence for a power series, we typically apply the ratio test. Calculus, Vol 2 2016-03-30T00:00:00Z Here we show that we can do the same thing for convergent power series. Calculus, Vol 2 2016-03-30T00:00:00Z Therefore, the power series diverges at x = 5. Calculus, Vol 2 2016-03-30T00:00:00Z Evaluating the derivative and indefinite integral in this way is called term-by-term differentiation of a power series and term-by-term integration of a power series, respectively. Calculus, Vol 2 2016-03-30T00:00:00Z The ability to differentiate and integrate power series term- by-term also allows us to use known power series representations to find power series representations for other functions. Calculus, Vol 2 2016-03-30T00:00:00Z First, we state Term-by-Term Differentiation and Integration for Power Series, which provides the main result regarding differentiation and integration of power series. Calculus, Vol 2 2016-03-30T00:00:00Z Note that although Term-by-Term Differentiation and Integration for Power Series guarantees the same radius of convergence when a power series is differentiated or integrated term-by-term, it says nothing about what happens at the endpoints. Calculus, Vol 2 2016-03-30T00:00:00Z Figure 6.3 The graph shows a function and three approximations of it by partial sums of a power series. Calculus, Vol 2 2016-03-30T00:00:00Z Up to this point, we have shown several techniques for finding power series representations for functions. Calculus, Vol 2 2016-03-30T00:00:00Z However, how do we know that these power series are unique? Calculus, Vol 2 2016-03-30T00:00:00Z This fact should not seem surprising if we think of power series as polynomials with an infinite number of terms. Calculus, Vol 2 2016-03-30T00:00:00Z At this point, however, we are still limited as to the functions for which we can find power series representations. Calculus, Vol 2 2016-03-30T00:00:00Z Next we consider functions involving an expression similar to the sum of a geometric series and show how to represent these functions using power series. Calculus, Vol 2 2016-03-30T00:00:00Z We will show in a later chapter that these series often arise when studying power series. Calculus, Vol 2 2016-03-30T00:00:00Z The ratio test will be especially useful in the discussion of power series in the next chapter. Calculus, Vol 2 2016-03-30T00:00:00Z So, the question is, when can we represent a function by a power series? Calculus, Vol 2 2016-03-30T00:00:00Z Use a power series to represent each of the following functions f . Calculus, Vol 2 2016-03-30T00:00:00Z In the preceding section on power series and functions we showed how to represent certain functions using power series. Calculus, Vol 2 2016-03-30T00:00:00Z Next, we show how to find power series representations for many more functions by introducing Taylor series. Calculus, Vol 2 2016-03-30T00:00:00Z In the following exercises, use partial fractions to find the power series of each function. Calculus, Vol 2 2016-03-30T00:00:00Z Earlier in the chapter, we showed how you could combine power series to create new power series. Calculus, Vol 2 2016-03-30T00:00:00Z The ability to differentiate power series term by term makes them a powerful tool for solving differential equations. Calculus, Vol 2 2016-03-30T00:00:00Z In this section we discuss how power series can be combined, differentiated, or integrated to create new power series. Calculus, Vol 2 2016-03-30T00:00:00Z Using Uniqueness of Power Series on the uniqueness of power series representations, we know that coefficients of the same degree must be equal. Calculus, Vol 2 2016-03-30T00:00:00Z We now show how to use power series to approximate this integral. Calculus, Vol 2 2016-03-30T00:00:00Z In addition, they allow us to define new functions as power series, thus providing us with a powerful tool for solving differential equations. Calculus, Vol 2 2016-03-30T00:00:00Z First, it allows us to find power series representations for certain elementary functions, by writing those functions in terms of functions with known power series. Calculus, Vol 2 2016-03-30T00:00:00Z Similar results hold for power series centered at x = a. Calculus, Vol 2 2016-03-30T00:00:00Z For simplicity, we state the theorem for power series centered at x = 0. Calculus, Vol 2 2016-03-30T00:00:00Z In Example 6.5, we showed how to find power series for certain functions. Calculus, Vol 2 2016-03-30T00:00:00Z We can view the present value as a power series in r, which converges as long as | 1 | < 1. Calculus, Vol 2 2016-03-30T00:00:00Z It is possible that the differentiated and integrated power series have different behavior at the endpoints than does the original series. Calculus, Vol 2 2016-03-30T00:00:00Z In the next example, we show how to use Combining Power Series and the power series for a function f to construct power series for functions related to f. Calculus, Vol 2 2016-03-30T00:00:00Z Term-by-Term Differentiation and Integration for Power Series does not guarantee anything about the behavior of this power series at the endpoints. Calculus, Vol 2 2016-03-30T00:00:00Z We then present two common applications of power series. Calculus, Vol 2 2016-03-30T00:00:00Z We also showed previously in this chapter how power series can be differentiated term by term to create a new power series. Calculus, Vol 2 2016-03-30T00:00:00Z Here we show how to solve it using power series. Calculus, Vol 2 2016-03-30T00:00:00Z Since we are combining these two power series, the interval of convergence of the difference must be the smaller of these two intervals. Calculus, Vol 2 2016-03-30T00:00:00Z If a function f has a power series representation at x = a, then it is given by its Taylor series at x = a. Calculus, Vol 2 2016-03-30T00:00:00Z In the previous two sections we discussed how to find power series representations for certain types of functions––specifically, functions related to geometric series. Calculus, Vol 2 2016-03-30T00:00:00Z Here we discuss power series representations for other types of functions. Calculus, Vol 2 2016-03-30T00:00:00Z In particular, we address the following questions: Which functions can be represented by power series and how do we find such representations? Calculus, Vol 2 2016-03-30T00:00:00Z In Example 6.6 we show how to do the opposite: given a power series, determine which function it represents. Calculus, Vol 2 2016-03-30T00:00:00Z This power series for f is known as the Taylor series for f at a. Calculus, Vol 2 2016-03-30T00:00:00Z Recall from Uniqueness of Power Series that power series representations are unique. Calculus, Vol 2 2016-03-30T00:00:00Z Therefore, if a function f has a power series at a, then it must be the Taylor series for f at a. Calculus, Vol 2 2016-03-30T00:00:00Z First, we show how power series can be used to solve differential equations. Calculus, Vol 2 2016-03-30T00:00:00Z Being able to multiply two power series provides another way of finding power series representations for functions. Calculus, Vol 2 2016-03-30T00:00:00Z Second, we show how power series can be used to evaluate integrals when the antiderivative of the integrand cannot be expressed in terms of elementary functions. Calculus, Vol 2 2016-03-30T00:00:00Z The power series converges for all real numbers x. Calculus, Vol 2 2016-03-30T00:00:00Z |
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