一种交错级数最小误差范围的余项估计

A Remainder Estimation of an Alternating Series with Minimum Error Range

研究了由奇自然数倒数组成的无穷交错级数余项估计的一个方法。给出了该级数第一项到前n项部分和余项的估计不等式,并进行了证明,估计公式极其简单。计算表明当n=5时余项估计的相对误差已小于1%,当n=16时相对误差小于1‰。

A new method is introduced about the remainder estimation of infinite alternating series constructed by the reciprocal of odd natural numbers. The inequality of remainder estimation range is given with the remainder of the first term to any the nth partial sum of the series and it is proved. Estimation formula is extremely simple. It is shown by calculation that relative error of the remainder estimation is smaller than 1% when n = 5 and smaller than 1‰ when n = 16.