limn→∞(1+1/3)(1+1/3^2)(1+3^4)…(1+3^2^n)rt

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limn→∞(1+1/3)(1+1/3^2)(1+3^4)…(1+3^2^n)rt
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limn→∞(1+1/3)(1+1/3^2)(1+3^4)…(1+3^2^n)rt
limn→∞(1+1/3)(1+1/3^2)(1+3^4)…(1+3^2^n)
rt

limn→∞(1+1/3)(1+1/3^2)(1+3^4)…(1+3^2^n)rt
limn→∞(1+1/3)(1+1/3^2)(1+1/3^4)…(1+1/3^n)
=(3/2)(1-1/3ⁿ)
=3/2

ln(1+x) &lt; x 对一切 x&gt; 0 成立84于是:ln((1+1&#47;2)(1+2&#47;4)(1+3&#47;8)(1+4&#47;16)……(1+n&#47;2^n))= ln(1+ 1&#47;2) + ....+ln(1+n&#47;2^n) &lt; 1&#47;2+...+ n&#47;2^n设 A = 1&#47;2+...+ n&#47;2^n32A = 1...

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ln(1+x) &lt; x 对一切 x&gt; 0 成立84于是:ln((1+1&#47;2)(1+2&#47;4)(1+3&#47;8)(1+4&#47;16)……(1+n&#47;2^n))= ln(1+ 1&#47;2) + ....+ln(1+n&#47;2^n) &lt; 1&#47;2+...+ n&#47;2^n设 A = 1&#47;2+...+ n&#47;2^n32A = 1 + 2&#47;2 + ... + n&#47;2^(n-1)两式相减 得:A = 1 +1/2 + 1/4 + ...+ 1/2^(n-1) - n/2^n= 2 - 1/2^(n-1) - n/2^n< 2所以: ln((1+1&#47;2)(1+2&#47;4)(1+3&#47;8)(1+4&#47;16)……(1+n&#47;2^n)) < 2===> (1+1&#47;2)(1+2&#47;4)(1+3&#47;8)(1+4&#47;16)……(1+n&#47;2^n) < e^2 < 9

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