Question

求极限 n->∞ lim n*((1+1/n)^n - e)

Answer 1

lim(n->∞) n[ ( 1+1/n)^n - e]
y=1/x
y->0
( 1+y)^(1/y)
=e^[(1/y)ln(1+y)]
=e^[(1/y) ( y - (1/2)y^2) ] +o(y)
=e^[ 1- (1/2)y ] +o(y)
( 1+y)^(1/y) - e
=e. ( e^[- (1/2)y ] -1 ) + o(y)
=e. [ 1- (1/2)y -1 ] + o(y)
=e. [ - (1/2)y ] + o(y)
/
lim(x->∞) x[ ( 1+1/x)^x - e]
=lim(y->0) [ ( 1+y)^(1/y) - e]/y
=lim(y->0) e. [ - (1/2)y ] /y
=-(1/2)e
=>
lim(n->∞) n[ ( 1+1/n)^n - e] = -(1/2)e