(1+1/n2)的n次方求极限

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(1+1/n2)的n次方求极限(1+1/n2)的n次方求极限(1+1/n2)的n次方求极限={(1+1/2n)^2n}0.5=e^0.5①lim(n->∞)(1+1/n^2)^n=lim(n->∞)(

(1+1/n2)的n次方求极限
(1+1/n2)的n次方求极限

(1+1/n2)的n次方求极限
={(1+1/2n)^2n}0.5=e^0.5


lim(n->∞) (1+1/n^2)^n
=lim(n->∞) (1+1/n^2)^[(n^2)/n]
=lim(n->∞) {(1+1/n^2)^(n^2) }^(1/n)
∵ lim(n->∞) (1+1/n^2)^(n^2) = e
lim(n->∞) (1/n) = 0
= e^0
= 1

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lim(n->∞) (1+1/n^2)^n
=lim(n->∞) (1+1/n^2)^[(n^2)/n]
=lim(n->∞) {(1+1/n^2)^(n^2) }^(1/n)
∵ lim(n->∞) (1+1/n^2)^(n^2) = e
lim(n->∞) (1/n) = 0
= e^0
= 1

lim(n->∞) (1+1/2n)^n
=lim(n->∞) (1+1/2n)^[(2n)/2]
=lim(n->∞) {(1+1/2n)^(2n) }^(1/2)
∵ lim(n->∞) (1+1/2n)^(2n) = e
= e^(1/2)
= √e

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