數(shù)學(xué)系英文論文翻譯-閱讀頁(yè)
2024-09-28 13:57本頁(yè)面
【正文】 ) ! ( 1 ) !xx nnnr x x xnn? ??????, Noting the value of any set x , xe is a fixed constant, while the series (956) is absolutely convergent, so the general when the item when n?? , 1 0( 1)!nxn? ?? , so when n → ∞, there 1 0( 1)!nx xen? ??, From this lim ( ) 0nn rx?? ? This indicates that the series (956) does converge to ()xf x e? , therefore 2111 2 ! !xne x x xn? ? ? ? ? ?… … ( x??? ??? ). Such use of Maclaurin formula are expanded in power series method, although the procedure is clear, but operators are often too Cumbersome, so it is generally more convenient to use the following power series expansion method. Prior to this, we have been a function x?11 , xe and sinx power series expansion, the use of these known expansion by power series of operations, we can achieve many functions of power series expansion. This demand function of power series expansion method is called indirect expansion. Example 2 Find the function ( ) cosf x x? , 0x? ,Department in the power series expansion. Solution because (sin ) cosxx?? , And 3 5 2 11 1 1s in ( 1 )3 ! 5 ! ( 2 1 ) !nnx x x x xn ?? ? ? ? ? ? ??… …，（ x??? ??? ） Therefore, the power series can be itemized according to the rules of derivation can be 3 4 21 1 1c o s 1 ( 1 )2 ! 4 ! ( 2 ) !nnx x x xn? ? ? ? ? ? ?… …,（ x??? ??? ） Third, the function power series expansion of the application example The application of power series expansion is extensive, for example, can use it to set some numerical or other approximate calculation of integral value. Example 3 Using the expansion to estimatearctanx the value of? . Solution because πarctan1 4? Because of 3 5 7a r c ta n 3 5 7x x xxx? ? ? ? ? …, ( 11x? ? ? ), So there 1 1 14 a r c ta n 1 4 (1 )3 5 7? ? ? ? ? ? ? … Available right end of the first n items of the series and as an approximation of ? . However, the convergence is very slow progression to get enough items to get more accurate estimates of ? value.
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