求数列极限(1+(1/n)+(1/n^2))^n

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求数列极限(1+(1/n)+(1/n^2))^n求数列极限(1+(1/n)+(1/n^2))^n求数列极限(1+(1/n)+(1/n^2))^n取对数得limnln(1+1/n+1/n²)令

求数列极限(1+(1/n)+(1/n^2))^n
求数列极限(1+(1/n)+(1/n^2))^n

求数列极限(1+(1/n)+(1/n^2))^n
取对数得
lim n ln (1+ 1/n + 1/n² )
令t = 1/n
= lim ln(1+t+t²) / t
= lim (1+2t)/(1+t+t²) 【罗比达法则】
= 1
∴lim (1+ 1/n + 1/n² ) ^ n = e