lim n[e2-(1+1/n)^2n],当n趋向于无穷
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lim n[e2-(1+1/n)^2n],当n趋向于无穷
答案等于e^2
答案等于e^2
lim(n→∞) n[e^2-(1+1/n)^(2n)]
=lim(n→∞) [e^2-(1+1/n)^(2n)]/(1/n)
注意(1+1/n)^(2n)的导数是2[ln(1+1/n)-1/(n+1)](1+1/n)^(2n)
原式=lim(n→∞) -2【[ln(1+1/n)-1/(n+1)](1+1/n)^(2n)】/(-1/n^2),用洛必达法则
-2[lim(n→∞) (1+1/n)^n]^2*lim(n→∞) [ln(1+1/n)-1/(n+1)]/(-1/n^2)
=-2e^2*lim(n→∞) [1/(n+1)-ln(1+1/n)]/(1/n^2),令t=1/n
=-2e^2*lim(t→0) [t/(1+t)-ln(1+t)]/t^2
=-2e^2*lim(t→0) [1/(1+t)^2-1/(1+t)]/(2t),再用洛必达法则
=-e^2*lim(t→0) 1/t*[1-(1+t)]/(1+t)^2
=-e^2*lim(t→0) -1/(1+t)^2
=e^2*1/(1+0)^2
=e^2
用粗略的方法:
展开(1+1/n)^(2n)=e^2-e^2/n+7e^2/(6n^2)-4e^2/(3n^3)+...
∴lim(n→∞) n[e^2-(1+1/n)^2n]
=lim(n→∞) n【e^2-[e^2-e^2/n+7e^2/(6n^2)-4e^2/(3n^3)+...]】
=lim(n→∞) n【e^2/n-7e^2/(6n^2)+4e^2/(3n^3)+...】
=lim(n→∞) 【e^2-7e^2/(6n)+4e^2/(3n^2)+...】
=e^2
=lim(n→∞) [e^2-(1+1/n)^(2n)]/(1/n)
注意(1+1/n)^(2n)的导数是2[ln(1+1/n)-1/(n+1)](1+1/n)^(2n)
原式=lim(n→∞) -2【[ln(1+1/n)-1/(n+1)](1+1/n)^(2n)】/(-1/n^2),用洛必达法则
-2[lim(n→∞) (1+1/n)^n]^2*lim(n→∞) [ln(1+1/n)-1/(n+1)]/(-1/n^2)
=-2e^2*lim(n→∞) [1/(n+1)-ln(1+1/n)]/(1/n^2),令t=1/n
=-2e^2*lim(t→0) [t/(1+t)-ln(1+t)]/t^2
=-2e^2*lim(t→0) [1/(1+t)^2-1/(1+t)]/(2t),再用洛必达法则
=-e^2*lim(t→0) 1/t*[1-(1+t)]/(1+t)^2
=-e^2*lim(t→0) -1/(1+t)^2
=e^2*1/(1+0)^2
=e^2
用粗略的方法:
展开(1+1/n)^(2n)=e^2-e^2/n+7e^2/(6n^2)-4e^2/(3n^3)+...
∴lim(n→∞) n[e^2-(1+1/n)^2n]
=lim(n→∞) n【e^2-[e^2-e^2/n+7e^2/(6n^2)-4e^2/(3n^3)+...]】
=lim(n→∞) n【e^2/n-7e^2/(6n^2)+4e^2/(3n^3)+...】
=lim(n→∞) 【e^2-7e^2/(6n)+4e^2/(3n^2)+...】
=e^2
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