﻿功能:正项级数求和

格式:Eulersum2(A)

说明:A是一个矩阵变量,其里面的数据按行依次存储交错级数由开始项到有限项n的数据.其实n就是矩阵A里元素的个数.对于n一般取多大这个没有统一的定论.一般n取不了多大就可以得到理想的结果.函数执行后返回级数的和.

原理:先把正项级数转换成交错级数,然后采用欧拉变换方法求交错级数的和

例子:

//求1+1/2^2+1/3^2+1/4^2+……=π^2/6=1.64493406684822

//操作命令如下
n =
[ 1.00000000000000  2.00000000000000  3.00000000000000  4.00000000000000  5.00000000000000
  6.00000000000000  7.00000000000000  8.00000000000000  9.00000000000000  10.0000000000000
  11.0000000000000  12.0000000000000  13.0000000000000  14.0000000000000  15.0000000000000
  16.0000000000000  17.0000000000000  18.0000000000000  19.0000000000000  20.0000000000000
  21.0000000000000  22.0000000000000  23.0000000000000  24.0000000000000  25.0000000000000
  26.0000000000000  27.0000000000000  28.0000000000000  29.0000000000000  30.0000000000000
  31.0000000000000  32.0000000000000  33.0000000000000  34.0000000000000  35.0000000000000
  36.0000000000000  37.0000000000000  38.0000000000000  39.0000000000000  40.0000000000000  ]
s=1/n^2//产生我们的数列,我们这里只取了前40项

s =
[ 1.00000000000000  0.25000000000000  0.11111111111111  0.06250000000000  0.04000000000000
  0.02777777777777  0.02040816326530  0.01562500000000  0.01234567901234  0.01000000000000
  0.00826446280991  0.00694444444444  0.00591715976331  0.00510204081632  0.00444444444444
  0.00390625000000  0.00346020761245  0.00308641975308  0.00277008310249  0.00250000000000
  0.00226757369614  0.00206611570247  0.00189035916824  0.00173611111111  0.00160000000000
  0.00147928994082  0.00137174211248  0.00127551020408  0.00118906064209  0.00111111111111
  0.00104058272632  0.00097656250000  0.00091827364554  0.00086505190311  0.00081632653061
  0.00077160493827  0.00073046018991  0.00069252077562  0.00065746219592  0.00062500000000  ]

eulersum2(s)//执行我们的求极限命令
ans  =
[  1.62054866190494  ]//发现和真实的解有一定差距,但是可以增加项数来增加精度

sum(s)//我们直接求和得到如下的效果,发现这个极限直接求和效果反而比欧拉变换好,但是有些函数却不是
ans =
[ 1.62024396300680 ]

//By 2012/3/17