>>5
区分求積法で解けるよ

lim ｛ 1/n + 1/(√n√(n+1)) + 1/(√n√(n+2)) + ･･････ + 1/(√n√(2n-1)) ｝
n→∞
=limΣ[k=0,n-1]( 1/(√n√(n+k))
n→∞
==lim(1/n)Σ[k=0,n-1]( 1/(√(1+k/n))
n→∞
=∫[0,1] (1/√(1+x)) dx
=2√2-2