L=lim(n->∞)[(2n)!/(n!.n^n)]^[1/(n+1)]

lnL

=lim(n->∞)[1/(n+1)]{∑(i:1->n)ln[(n+i)/n]}

=lim(n->∞)[1/(n+1)]{∑(i:1->n)ln[(1+i/n)}

=∫(0->1)ln(1+x)dx

=[xln(1+x)]、(0->1)-∫(0->1)x/(1+x)dx

=ln2-∫(0->1)[1-1/(1+x)]dx

=ln2-[x-ln、1+x、]、(0->1)

=ln2-(1-ln2)

=2ln2-1

L=e^(2ln2-1)=4e^(-1)

lim(n->∞)[(2n)!/(n!.n^n)]^[1/(n+1)]=4e^(-1)