數(shù)學(xué)實驗之pi的近似計算

1、1/14/20221/14/2022 在本次試驗中，我們將追溯關(guān)于圓周率的計算歷程。通過對割圓術(shù)、韋達(dá)公式、級數(shù)加速法、迭代法等計算方法的介紹和計算體驗，感受數(shù)學(xué)思想和數(shù)學(xué)方法的發(fā)展過程，提高對極限和級數(shù)收斂性及收斂速度的綜合認(rèn)識，同時使我們看到數(shù)學(xué)家對科學(xué)真理的永無止境的追求。1/14/2022主要內(nèi)容 四、利用級數(shù)計算二、韋達(dá)(VieTa)公式 三、數(shù)值積分方法 一、割圓術(shù)六、拉馬努金(Ramanujan)公式五、蒙特卡羅(Monte Carlo)法He Renbin1/14/2022He Renbin1/14/2022He Renbin1/14/2022He Renbin1/14/202

2、2He Renbin1/14/202222 . 62 . 6421nnaaHe Renbin1/14/202226)1(642nnaSnnnSSSS222He Renbin1/14/2022He Renbin1/14/2022He Renbin1/14/2022He Renbin1/14/20222222 222 222He Renbin1/14/2022)8sin()8cos()4cos()2cos(8)4sin()4cos()2cos(4)2sin()2cos(2sinttttttttttHe Renbin1/14/2022）得到取）有令11112cos(2,22cos(sin,N)2co

3、s()2sin(2sinnnnnnNnNNtttttttttHe Renbin1/14/20222222122214cos)8cos(22)4cos(He Renbin1/14/2022He Renbin1/14/202210214dxxHe Renbin1/14/20221n0i1iii2)f(x)f(xn1S起來就得到將所有這所有這些梯形。n,0,1,ii/n,x等分，即n將積積分區(qū)He Renbin1/14/202242)(61)4(1)(121110211niiniiniiiiiiyyyynSyyynSxfy令He Renbin1/14/2022012) 1(4kkkHe Renbin

4、1/14/202212112) 1(4)(0nknrnkkHe Renbin1/14/2022022122) 12(1816kkkkHe Renbin1/14/2022001212) 1(4，可得1取12) 1(arctankkkkkkxkxx時時He Renbin1/14/202211191202tan12tan24tan12512tan1tan22tan51arctan,51tan222xxxHe Renbin1/14/20221201202391 12) 1(451 12) 1(162391arctan451arctan1641623914tan114tantankkkkkkkkHe R

5、enbin1/14/2022He Renbin1/14/2022He Renbin1/14/2022nmSSnm4,41He Renbin1/14/2022He Renbin1/14/2022mdnl2He Renbin1/14/2022mdnl2He Renbin1/14/2022044396263901103 ) !()!4(9801221nnnnnHe Renbin1/14/2022He Renbin1/14/20220332364032054514013413591409 )!3() !()!6() 1(640320121nnnnnnnHe Renbin1/14/2022)1 (2)1 (246111122121404410nnnnnnnnnnnnyyyayaazzyyzyHe Renbin1/14/2022neann421641He Renbin1/14/2022)52(2552/1,)/2/21 (25)73)7(),15()52(),25(5121121112102551321210nnnnnnnnnnnnnnnnnnnnnnyyyyayaaeedyycdcdeydycyHe Renbin1/14/2022neann51