Codeforces #454 C | 哈喽哈咯

Written by razrlele 20:42 January 15, 2015

Twilight Sparkle was playing Ludo with her friends Rainbow Dash, Apple Jack and Flutter Shy. But she kept losing. Having returned to the castle, Twilight Sparkle became interested in the dice that were used in the game.

The dice has m faces: the first face of the dice contains a dot, the second one contains two dots, and so on, the m-th face contains mdots. Twilight Sparkle is sure that when the dice is tossed, each face appears with probability $\text{[math]}$ . Also she knows that each toss is independent from others. Help her to calculate the expected maximum number of dots she could get after tossing the dice n times.

Input

A single line contains two integers m and n (1 ≤ m, n ≤ 10⁵).

Output

Output a single real number corresponding to the expected maximum. The answer will be considered correct if its relative or absolute error doesn’t exceed 10 ^- 4.

Sample test(s)

input

6 1

output

1	3.500000000000

input

6 3

output

1	4.958333333333

input

2 2

output

1	1.750000000000

Note

Consider the third test example. If you’ve made two tosses:

You can get 1 in the first toss, and 2 in the second. Maximum equals to 2.
You can get 1 in the first toss, and 1 in the second. Maximum equals to 1.
You can get 2 in the first toss, and 1 in the second. Maximum equals to 2.
You can get 2 in the first toss, and 2 in the second. Maximum equals to 2.

The probability of each outcome is 0.25, that is expectation equals to:

$\text{[math]}$ You can read about expectation using the following link: http://en.wikipedia.org/wiki/Expected_value

我觉得应该算是数论的题目?

关键在于找规律, 比如输入为 6 2

遍历所有的输出

6 6 6 6 6 6

5 5 5 5 5 6

4 4 4 4 5 6

3 3 3 4 5 6

2 2 3 4 5 6

1 2 3 4 5 6

从此可以发现最终的输出其实是一个棱长为m的n方体, 一层一层向外面包, 故第i层的元素个数即为 i^n-(i-1)^n, 然后第i层的元素都为i, 然后总的元素个数是m^n, 注意最后求的时候不能直接算m^n, 不然会溢出, 换成算(i/m)^n即可.

/********************************************************

* File Name: 454C.cpp

* Author: razrLeLe

* Mail: razrlele@outlook.com

* Homepage: https://yueyu.io

* Created Time: Thu 15 Jan 2015 06:14:05 PM CST

******************************************************/

#include "vector"

#include "set"

#include "deque"

#include "queue"

#include "algorithm"

#include "functional"

#include "iostream"

#include "cstdio"

#include "cmath"

#include "cstdlib"

#include "string"

#include "cstring"

#include "string.h"

#include "map"

#include "cctype"

#include "list"

#include "stack"

using namespace std;

#define INF 0x3f3f3f3f

#define LOCAL

#define REP(i, start, end) for(i = (start); i <= (end); ++i)

double ans;

int m, n;

int main()

{

#ifdef LOCAL

freopen("/home/razrlele/build/data.txt", "r", stdin);

freopen("/home/razrlele/build/out.txt", "w", stdout);

#endif

double i;

while(cin >> m >> n)

{

ans = 0;

REP(i, 1, m)

ans += i*(pow(i/m, n) - pow((i-1)/m, n));

printf("%.12lf\n", ans);

}

return 0;

}