Sum of series 1*1! + 2*2! + ……..+ n*n!

Given n, we need to find sum of 1*1! + 2*2! + ……..+ n*n!

Examples:

Input: 1
Output: 1

Input: 3
Output: 23
1 * 1! + 2 * 2! + 3 * 3! = 1 + 4 + 18 = 23

We may assume that overflow does not happen.

A simple solution is to compute terms one by one and add to result.

An efficient solution is based on direct formula (n + 1)! – 1

How does this formula work?

We basically need to compute below sum.
∑(i * i!) Where i varies from 1 to n
= ∑((i + 1 – 1) * i!)
= ∑((i+1) * i!) – ∑i!
= ∑(i + 1)! – ∑(i!)

∑(i + 1)! = 2! + 3! + … (n+1)! where 1 <= i <= n —–(1)
∑(i!) = 1! + 2! + 3! + … (n)! where 1 <= i <= n —–(2)

Subtracting second from first, we get (n+1)! – 1

C++

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// CPP program to find sum of the series.
#include <bits/stdc++.h>
using namespace std;
  
int factorial(int n)
{
    int res = 1;
    for (int i = 2; i <= n; i++)
        res = res * i;
    return res;
}
  
// Function to calculate required series
int calculateSeries(int n)
{
    return factorial(n + 1) - 1;
}
  
// Drivers code
int main()
{
    int n = 3;
    cout << calculateSeries(n);
    return 0;
}

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Java

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// Java program to find sum of 
// the series.
import java.io.*;
  
class GFG {
  
    static int factorial(int n)
    {
        int res = 1;
        for (int i = 2; i <= n; i++)
            res = res * i;
        return res;
    }
      
    // Function to calculate required
    // series
    static int calculateSeries(int n)
    {
        return factorial(n + 1) - 1;
    }
      
    // Drivers code
    public static void main (String[] args)
    {
        int n = 3;
        System.out.println( 
                       calculateSeries(n));
    }
}
  
// This code is contributed by anuj_67.

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Python3

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# Python program to find sum of 
# the series.
  
def factorial(n):
    res = 1
    for i in range(2, n+1):
        res = res * i
    return res
  
# Function to calculate required
# series
def calculateSeries(n):
    return factorial(n + 1) - 1
  
# Drivers code
n = 3
print(calculateSeries(n))
  
# This code is contributed by 
# Ansu Kumari.

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C#

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// C# program to find 
// sum of the series.
using System;
  
class GFG
{
    static int factorial(int n)
    {
        int res = 1;
        for (int i = 2; i <= n; i++)
            res = res * i;
        return res;
    }
      
    // Function to calculate 
    // required series
    static int calculateSeries(int n)
    {
        return factorial(n + 1) - 1;
    }
      
    // Driver code
    static public void Main ()
    {
        int n = 3;
        Console.WriteLine( 
                calculateSeries(n));
    }
}
  
// This code is contributed by ajit.

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PHP

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<?php
// PHP program to find
// sum of the series.
  
function factorial($n)
{
    $res = 1;
    for ($i = 2; $i <= $n; $i++)
        $res = $res * $i;
    return $res;
}
  
// Function to calculate
// required series
function calculateSeries($n)
{
    return factorial($n + 1) - 1;
}
  
// Driver code
$n = 3;
echo calculateSeries($n);
  
// This code is contributed 
// by akt_mit
?>

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Output:

23


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