sum_{k=10}^{100000}1/(0.4187*k-3.1557)^3.03=1.649447192
sum_{k=10}^{200000}1/(0.4187*k-3.1557)^3.03=1.649447637
sum_{k=10}^{500000}1/(0.4187*k-3.1557)^3.03=1.649447638
有了公式及计算软件,就很容易得到结果了.... 由余项公式: (p=30)
|R|
数学星空 发表于 2012-2-15 22:41 http://bbs.emath.ac.cn/images/common/back.gif
和普通级数不同，渐近级数直接求和的话不是项数越多越好。另外你在6楼的积分算错了。 多谢楼上的提醒!
我们若取前10项可以得到:
a = 0.4187,b = -3.1557,a_1 = 2.4631,a_n = n-7.5369,m = 3.03,t = 10
sum_{k=10}^{n}1/(0.4187*k-3.1557)^m
=1/a^m*(1/(m-1)*(1/a_1^(m-1)-1/a_n^(m-1))+1/(2*a_n^m)-sum_{k=1}^{t-1}(prod_{j=0}^{2*k-2}(m+j)*(1/a_n^(m+2*k-1)*B_(2*k)/((2*k)!))))+C+R
=1.105170899-6.888711398/(n-7.5369)^2.03+6.992042070/(n-7.5369)^3.03-3.530981244/(n-7.5369)^4.03+1.192936128/(n-7.5369)^6.03-
1.204038955/(n-7.5369)^8.03+2.182648707/(n-7.5369)^10.03-6.097686636/(n-7.5369)^12.03+24.19304568/(n-7.5369)^14.03-
129.2013937/(n-7.5369)^16.03+893.3788567/(n-7.5369)^18.03-7764.356527/(n-7.5369)^20.03+C+R
注: C=1/a^m*(lim_{n->oo}(sum_{k=10}^{n}1/(0.4187*k-3.1557)^m-1/(m-1)*(1/a_1^(m-1)-1/a_n^(m-1))))=0.54427685
余项|R|=O(1/a_{n}^{m+2*k-1})=O(1/(n-7.5369)^20.03)
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