∑1÷n³的敛散性

∑1÷n³的敛散性
∑1÷n³的敛散性

∑1÷n³的敛散性
利用柯西审敛原理!
对任意的正整数p,
|U（n+1）+U(n+2)+……U（n+p）|=1/(n+1)^3+1/(n+2)^3+……+1/(n+p)^3|<1/(n+1)^2+1/(n+2)^2+……1/(n+p)^2<1/n*(n+1)+1/(n+1)*(n+2)+……1/(n+p-1)*(n+p)=(1/n-1/(n+1))+(1/(n+1)-1/(n+2))+……(1/(n+p-1)-1/(n+p))=1/n-1/(n+p)<1/n
所以对于任意给定正数a,取正整数N>=1/a,则当n>N时,对于任意正整数p,都有
|U(n+1)+U(n+2)+…U（n+p)|依柯西审敛原理,命题得证