利用极限存在准则证明limn→∞[1n2+1+1n2+2+…+1n2+n]=1．

Question

利用极限存在准则证明

lim

n→∞

[(1)/(√(n2+1))+(1)/(√(n2+2))+…+(1)/(√(n2+n))]=1．

Answer 1

证明：∵(1)/(n+1)＜(1)/(√(n2+n))＜(1)/(n)，
∴(1)/(n+1)+(1)/(n+1)+…+(1)/(n+1)＜(1)/(√(n2+1))+(1)/(√(n2+2))+…+(1)/(√(n2+n))＜(1)/(n)+(1)/(n)+…+(1)/(n)
即有(n)/(n+1)＜(1)/(√(n2+1))+(1)/(√(n2+2))+…+(1)/(√(n2+n))＜1．
由于

lim

n→∞

(n)/(n+1)=1，

lim

n→∞

1=1，
则有

lim

n→∞

[(1)/(√(n2+1))+(1)/(√(n2+2))+…+(1)/(√(n2+n))]=1．