「刷题记录」 [SHOI2002] 百事世界杯之旅

第一道有关极限期望的数学题，记录一下。

我们设 $f_i$ 是凑齐前 $i$ 个球星期望需要买的饮料数。

$$E = 1 \times \dfrac{n - i}{n} + 2 \times \dfrac{i}{n} \times \dfrac{n - i}{n} + 3 \times \left ( \dfrac{i}{n} \right ) ^ 2 \times \dfrac{n - i}{n} + 4 \times \left ( \dfrac{i}{n} \right ) ^ 3 \times \dfrac{n - i}{n} \dots + k \times \left( \dfrac{i}{n} \right ) ^ {k - 1} \times \dfrac{n - i}{n}$$

其中 $k$ 是一个极大数，可以看作是无限，我们由此可以推出下面的式子：

$$\dfrac{i}{n} E = \dfrac{i}{n} \times \dfrac{n - i}{n} + 2 \times \left ( \dfrac{i}{n} \right ) ^ 2 \times \dfrac{n - i}{n} + 3 \times \left ( \dfrac{i}{n} \right ) ^ 3 \times \dfrac{n - i}{n} + 4 \times \left ( \dfrac{i}{n} \right ) ^ 4 \times \dfrac{n - i}{n} \dots + k \times \left( \dfrac{i}{n} \right ) ^ {k} \times \dfrac{n - i}{n}$$

相减得：

$$\dfrac{n - i}{n} E = \dfrac{n - i}{n} + \dfrac{i}{n} \cdot \dfrac{n - i}{n} + \left ( \dfrac{i}{n} \right ) ^ 2 \cdot \dfrac{n - i}{n} + \dots + \left ( \dfrac{i}{n} \right ) ^ {k - 1} \cdot \dfrac{n - i}{n} - k \cdot \left ( \dfrac{i}{n} \right ) ^ k \cdot \dfrac{n - i}{n}$$

由于 $k$ 为无限大，所以 $k \cdot \left ( \dfrac{i}{n} \right ) ^ k \cdot \dfrac{n - i}{n}$ 就无限接近于 $0$，所以它对答案的影响也无限接近于 $0$，我们可以直接将其省略，那么期望式子就变成这样了：

$$\dfrac{n - i}{n} E = \dfrac{n - i}{n} + \dfrac{i}{n} \cdot \dfrac{n - i}{n} + \left ( \dfrac{i}{n} \right ) ^ 2 \cdot \dfrac{n - i}{n} + \dots + \left ( \dfrac{i}{n} \right ) ^ {k - 1} \cdot \dfrac{n - i}{n}$$

将系数化为 $1$ 得：

$$E = 1 + \dfrac{i}{n} + \left ( \dfrac{i}{n} \right ) ^ 2 + \dots + \left ( \dfrac{i}{n} \right ) ^ {k - 1}$$

这里我们可以用等比数列求和公式来做。设 $S = \dfrac{i}{n} + \left ( \dfrac{i}{n} \right ) ^ 2 + \dots + \left ( \dfrac{i}{n} \right ) ^ {k - 1}$，则 $\dfrac{i}{n} S = \left ( \dfrac{i}{n} \right ) ^ 2 + \left ( \dfrac{i}{n} \right ) ^ 3 + \dots + \left ( \dfrac{i}{n} \right ) ^ {k}$

$$\dfrac{n - i}{n} S = S - \dfrac{i}{n} S = \dfrac{i}{n} - \left ( \dfrac{i}{n} \right ) ^ {k}$$

还是考虑极限，$k$ 为无限大，则 $\left ( \dfrac{i}{n} \right ) ^ {k}$ 无限接近于 $0$，都答案的影响可以忽略不计，所以，

$$\dfrac{n - i}{n} S \approx \dfrac{i}{n}\\ S \approx \dfrac{i}{n - i}\\ E = 1 + S = \dfrac{n}{n - i}\\ $$

求出了期望，我们可以写出递推式：$f_{i + 1} = f_i + \dfrac{n}{n - i}$，转化一下，就是 $f_i = f_{i - 1} + \dfrac{n}{n - (i - 1)}$。

现在，我们可以一步一步的递推过去，也可以将 $f_i$ 的式子拆开，最后得到 $f_n = \dfrac{n}{n} + \dfrac{n}{n - 1} + \dots + \dfrac{n}{1} = n \times (\dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5} + \dots + \dfrac{1}{n})$，两种方法都可以。

//The code was written by yifan, and yifan is neutral!!!

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define bug puts("NOIP rp ++!");
#define rep(i, a, b, c) for (int i = (a); i <= (b); i += (c))
#define per(i, a, b, c) for (int i = (a); i >= (b); i -= (c))

template<typename T>
inline T read() {
    T x = 0;
    bool fg = 0;
    char ch = getchar();
    while (ch < '0' || ch > '9') {
        fg |= (ch == '-');
        ch = getchar();
    }
    while (ch >= '0' && ch <= '9') {
        x = (x << 3) + (x << 1) + (ch ^ 48);
        ch = getchar();
    }
    return fg ? ~x + 1 : x;
}

using fs = tuple<ll, ll>;

int n;

ll gcd(ll x, ll y) {
    if (y == 0) {
        return x;
    }
    return gcd(y, x % y);
}

fs operator+ (const fs& a, const fs& b) {
    ll g = gcd(get<1>(a), get<1>(b));
    ll lcm = get<1>(a) / g * get<1>(b);
    ll fza = get<0>(a), fzb = get<0>(b);
    fza *= (lcm / get<1>(a));
    fzb *= (lcm / get<1>(b));
    return fs(fza + fzb, lcm);
}

int main() {
    n = read<int>();
    fs res = fs(1, 1);
    rep (i, 2, n, 1) {
        res = res + fs(1, i);
    }
    res = fs(get<0>(res) * n, get<1>(res));
    ll num = get<0>(res) / get<1>(res);
    if (get<0>(res) % get<1>(res) == 0) {
        cout << num << '\n';
        return 0;
    }
    res = fs(get<0>(res) % get<1>(res), get<1>(res));
    int digitfz = 0, digitfm = 0, digitps = 0;
    ll tmpfz = get<0>(res), tmpfm = get<1>(res), tmpps = num;
    ll gc = gcd(tmpfz, tmpfm);
    res = fs(get<0>(res) / gc, get<1>(res) / gc);
    tmpfz = get<0>(res), tmpfm = get<1>(res), tmpps = num;
    while (tmpfz) {
        tmpfz /= 10;
        ++ digitfz;
    }
    while (tmpfm) {
        tmpfm /= 10;
        ++ digitfm;
    }
    while (tmpps) {
        tmpps /= 10;
        ++ digitps;
    }
    int maxx = max(digitfz, digitfm);
    rep (i, 1, digitps, 1) {
        cout << ' ';
    }
    cout << get<0>(res) << '\n';
    cout << num;
    rep (i, 1, maxx, 1) {
        cout << '-';
    }
    putchar('\n');
    rep (i, 1, digitps, 1) {
        cout << ' ';
    }
    cout << get<1>(res) << '\n';
    return 0;
}