※ 引述《MITjuching (CP3亞洲代言人)》之銘言:
: 化簡
: 1/2 + 3/2^2 + 5/2^3 + …………2n-1/2^n
: 拜託各位幫忙了
: 答案先奉上
: 3-[(2n+3)/2^n]
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: Sent from JPTT on my HUAWEI Che2-L12.
另 S_n = 1/2 + 3/2^2 + ... + (2n-1)/2^n
1/2 * (2k-1)/2^k = [(2*(k+1)-1)-2]/2^(k+1)
把 S_n 乘以 1/2 後得
1/2 S_n = (S_{n+1} - 1/2) - 2 * ( 1/2^2 + 1/2^3 + ... + 1/2^(n+1) )
= S_{n+1} - 1/2 - (1 - 2^(-n))
= S_n + (2n+3)/2^(n+1) - 3/2
所以 S_n = 3 - (2n+3)/2^n
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