作者Annihilator (> No LOVE (%))
看板Math
标题Re: [中学] 几题数列级数和指对数请教
时间Sat Apr 23 19:29:54 2011
※ 引述《izzylf (风)》之铭言：
: 各位版友好
: 这里有几题数列级数和指对数问题请教
: 因为符号不好在bbs显现,所以用printscreen的
: 图贴在自己的相簿
: 谢谢帮我回答的版友
: http://www.wretch.cc/album/show.php?i=izzylf0612&b=3&f=1619378913&p=1
1. S = 1 + (1+2) + (1+2+2^2) + ... + [1+2+2^2+...+2^(n-1)]
n-1 n-1
= Σ[2^(k+1)-1] = Σ2^(k+1) - n = 2^(n+1) - 2 - n
k=0 k=0
2. S_n = 1/1 + 1/(1+2) + ... + 1/(1+2+...+n)
n n
= Σ2/[(k+1)k] = 2Σ[1/k - 1/(k+1)] = 2 - 2/(n+1) → 2 as n→∞
k=1 k=1
|S-S_n|<0.01 => 2/(n+1)<0.01 => n > 2/0.01 - 1 = 199
n最小值=200
3. a = log_60(3), b = log_60(5)
[1-a-b]/[2-2b] = [1-log_60(3)-log_60(5)]/[2-2log_60(5)]
= log_60[60/(3x5)]/log_60[3600/25]
= log_60(4)/log_60(144)
= log_12(4)/log_12(144) = log_12(2)
Thus, 12^[(1-a-b)/(2-2b)] = 2
4. a^2 - 1 = (1/4)[47^(2/2010)+47^(-2/2010)]+1/2-1
= {(1/2)[47^(1/2010)-47^(-1/2010)]}^2
Then, Ans = [47^(-1/2010)]^2010 = 1/47
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1F：推 izzylf :感谢感谢!! 04/23 21:53