計算 lim n→無限[1/(n^(1/2))*1/((n+1)^(1/2))+1/(n^(1/2))*1/((n+2)^(1/2))+......+1/(n^(1/2))*1/((n+n)^(1/2))] 怎麼算?

計算lim(n->∞)_ [1/(n+1)^(1/2)+1/(n+2)^(1/2)+......+1/(n+n)^(1/2)]/n^(1/2) 怎麼算?
Sol
lim(n->∞)_ [1/(n+1)^(1/2)+1/(n+2)^(1/2)+......+1/(n+n)^(1/2)]/n^(1/2)
=lim(n->∞)_Σ(k=1 to n)_1/[(n+k)^(1/2)*n^(1/2)]
=lim(n->∞)_Σ(k=1 to n)_1/[(1+k/n)^(1/2)]/n
=∫(0 to 1)_1/(1+x)^(1/2)dx
=∫(0 to 1)_(1+x)^(－1/2)d(1+x)
=2(1+x)^(1/2)｜(0 to 1)
=2√2－2