数学外文翻译-------幂级数的展开及其应用.docVIP

Power Series Expansion and Its Applications 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. 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