例如,A(x)=1+2*x-26*x^2/2!+366*x^3/3!-6270*x^4/4!+99922*x^5/5!-1630730*x^6/6!-33526706*x ^ 7/7!+1685562866*x^8/8!-390576999182*x ^ 9/9!+。。。
哪里
1+4*x=经验(-1)*(1+A(x)+A(x)^4/2!+A(x)^9/3!+A(x)^16/4!+A(x)^25/5!+A(x)^36/6!+A(x)^49/7!+…+A(x)^(n^2)/n!+…)。
相关表格。
系数表x^k/k!在A(x)^(n^2)中开始:
n=0:[1,0,0,0,0,…];
n=1:[1,2,-26,366,-6270,99922,-1630730,…];
n=2:[1,8,-56,-216,19800,-706472,14847688,…];
n=3:[1,18,54,-3906,34290,1326978,-99273402,…];
n=4:[1,32,544,-4704,-308640,6962272,154469920,…];
n=5:[1,50,1750,25950,-936750,-37790750,1459186150,…];
n=6:[1,72,4104,159336,1906200,-19221928,-7838021880,…];
n=7:[1,98,8134,535374,23730210,239390578,-52296366122,…]。。。
其中沿列的无限项之和可通过以下方式表示:
1=(1+1+1/2!+1/3!+1/4!+1/5!+…)/e;
4=(0+2+8/2!+18/3!+32/4!+50/5!+…)/e;
0 = (0 + -26 + -56/2! + 54/3! + 544/4! + 1750/5! + ...);
0 = (0 + 366 + -216/2! + -3906/3! + -4704/4! + 25950/5! + ...);
0 = (0 + -6270 + 19800/2! + 34290/3! + -308640/4! + -936750/5! + ...);
0 = (0 + 99922 + -706472/2! + 1326978/3! + 6962272/4! + -37790750/5! ...); ...
并可用于确定此序列的所有项。
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