Question

数学极限计算题，4题。最好有过程，非常感谢。！

Answer 1

limn^2{[1+1/(n+1)]^(n+1)-(1+1/n)^n}

对括号内：

[1+1/(n+1)]^(n+1)-(1+1/n)^n

=e^{(n+1)ln[1+1/(n+1)]}-e^nln(1+1/n)

=e^nln(1+1/n)*｛e^{[(n+1)ln(1+1/(n+1))]-nln(1+1/n)}-1｝

有e^x的展开式e^x=1+x+o(x^2)

原式 =lim(n-∞)n^2*[1+1/(1+n)]^n*{(n+1)ln[1+1/(n+1)]-nln(1+1/n)}

=elim(n-∞)n^2{(n+1)ln[1+1/(n+1)]-nln(1+1/n)}

t=1/n =elim(t-0) [(1+1/t)ln(1+2t)/(1+t)-(1/t)ln(1+t)]/t^2

=elim(t-0) [(1+t)ln(1+2t)/(1+t)-ln(1+t)]/t^3

=elim(t-0) [(ln(1+2t)-2ln(1+t)+tln(1+2t)-tln(1+t)]/t^3

洛必达=elim(t-0).....

洛必达=elim(t-0).....

=elim(t-0)(4t+3)/[6(1+t)^2(1+2t)^2]

=e/2