A Remainder Estimation of an Alternating Series with Minimum Error Range

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.