※ 引述《changkh (月光華華)》之銘言：
: 最近我朋友碰上了一題數學題不知道怎麼解，不知道有沒有人學過呢？
: In the zipper model for the helix-coil transition, the partition
: function Q is give by:
: N
: Q= 1 + sigma * sum (N - i + 1) s^i
: i=1
: sum is summation.
: Where N is the number of residues in the chain.
: a) using the fact that
: N
: sum s^i = (s^(N+1) - S) / s - 1
: i=1
: show that the partition function in zipper model can be
: evaluated to give:
: Q= 1 + (sigma * s) * (s^(N+1) - (N+1)s + N) / (s - 1)^2
: b) show that the average fraction of helical residue, theta,
: can be obtained from
: theta = 1 / N * (ln Q對ln s的偏微分)
: 謝謝啦。
其實後來我也算出了a)
這題重點是
N
sum i*s^i怎麼算
i=1
我的算法是設
N N s^(N+1) - s
I=sum i*s^i 又令 J=sum s^i = -------------
i=1 i=1 s - 1
I = s + 2s^2 + 3s^3 + ... + (N-1)s^(N-1) + Ns^N
J= s + s^2 + s^3 + ... + S^(N-1) + s^N
則I-J= 0 + s^2 + 2s^3 + ... + (N-2)s^(N-1) + (N-1)s^N = I*s-Ns^(N+1)
所以I-J= Is - Ns^(N+1)
化簡可得I
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