Efficient Program to Compute Sum of Series 1/1! + 1/2! + 1/3! + 1/4! + .. + 1/n!

Given a positive integer n, write a function to compute the sum of the series 1/1! + 1/2! + .. + 1/n!
A Simple Solution is to initialize the sum as 0, then run a loop and call the factorial function inside the loop. 
Following is the implementation of a simple solution. 

C++

// A simple C++ program to compute sum of series 1/1! + 1/2! + .. + 1/n!
#include <iostream>
using namespace std;

//  Utility function to find
int factorial(int n)
{
    int res = 1;
    for (int i=2; i<=n; i++)
       res *= i;
    return res;
}

// A Simple Function to return value of 1/1! + 1/2! + .. + 1/n!
double sum(int n)
{
    double sum = 0;
    for (int i = 1; i <= n; i++)
        sum += 1.0/factorial(i);
    return sum;
}

// Driver program to test above functions
int main()
{
    int n = 5;
    cout << sum(n);
    return 0;
}

Java

// A simple Java program to compute
// sum of series 1/1! + 1/2! + .. + 1/n!
import java.io.*;

class GFG {

    // Utility function to find
    static int factorial(int n)
    {
        int res = 1;
        for (int i = 2; i <= n; i++)
        res *= i;
        return res;
    }

    // A Simple Function to return value
    // of 1/1! + 1/2! + .. + 1/n!
    static double sum(int n)
    {
        double sum = 0;
        for (int i = 1; i <= n; i++)
            sum += 1.0/factorial(i);
        return sum;
    }

    // Driver program
    public static void main (String[] args)
    {
        int n = 5;
        System.out.println(sum(n));
    }
}

// This code is contributed by Ajit.

Python3

# Python3 program to compute sum of series
# 1/1! + 1/2! + .. + 1/n!

# Function to find factorial of a number
def factorial(n):
    res = 1
    for i in range(2, n + 1):
            res *= i
    return res

# A Simple Function to return value
# of 1/1! + 1/2! + .. + 1/n!
def sum(n):
    s = 0.0

    for i in range(1, n + 1):
        s += 1.0 / factorial(i)
    print(s)

# Driver program to test above functions
n = 5
sum(n)

# This code is contributed by Danish Raza

C#

// A simple C# program to compute sum
// of series 1/1! + 1/2! + .. + 1/n!
using System;

class GFG {

    // Utility function to find
    static int factorial(int n)
    {
        int res = 1;
        for (int i = 2; i <= n; i++)
            res *= i;

        return res;
    }

    // A Simple Function to return value
    // of 1/1! + 1/2! + .. + 1/n!
    static double sum(int n)
    {
        double sum = 0;
        for (int i = 1; i <= n; i++)
            sum += 1.0/factorial(i);

        return sum;
    }

    // Driver program
    public static void Main ()
    {
        int n = 5;

        Console.WriteLine(sum(n));
    }
}

// This code is contributed by Sam007.

PHP

<?php
// A simple PHP program to compute
// sum of series 1/1! + 1/2! + .. + 1/n!

// Utility function to find
function factorial($n)
{
    $res = 1;
    for ($i = 2; $i <= $n; $i++)
        $res *= $i;
    return $res;
}

// A Simple Function to return
// value of 1/1! + 1/2! + .. + 1/n!
function sum($n)
{
    $sum = 0;
    for ($i = 1; $i <= $n; $i++)
        $sum += 1.0 / factorial($i);
    return $sum;
}

// Driver Code
$n = 5;
echo(sum($n));

// This code is contributed by Ajit.
?>

Javascript

<script>

//Javascript program to compute
// sum of series 1/1! + 1/2! + .. + 1/n!

// Utility function to find
function factorial(n)
{
    let res = 1;
    for (let i = 2;  i <= n; i++)
        res *= i;
    return res;
}

// A Simple Function to return
// value of 1/1! + 1/2! + .. + 1/n!
function sum(n)
{
    let sum = 0;
    for (let i = 1; i <= n; i++)
        sum += 1.0 / factorial(i);
    return sum;
}

// Driver Code

let n = 5;
document.write(sum(n).toFixed(5));

// This code is contributed by sravan kumar

</script>

Output: 

1.71667

Time complexity: O(n * n)

Auxiliary Space: O(1), since no extra space has been taken.
An Efficient Solution can find the sum in O(n) time. The idea is to calculate factorial in the same loop as the sum. Following is the implementation of this idea. 

C++

// A simple C++ program to compute sum of series 1/1! + 1/2! + .. + 1/n!
#include <iostream>
using namespace std;

// An Efficient Function to return value of 1/1! + 1/2! + .. + 1/n!
double sum(int n)
{
    double sum = 0;
    int fact = 1;
    for (int i = 1; i <= n; i++)
    {
       fact *= i;         // Update factorial
       sum += 1.0/fact;   // Update series sum
    }
    return sum;
}

// Driver program to test above functions
int main()
{
    int n = 5;
    cout << sum(n);
    return 0;
}

Java

// A simple Java program to compute
// sum of series 1/1! + 1/2! + .. + 1/n!
import java.io.*;

class GFG {

    // An Efficient Function to return
    // value of 1/1! + 1/2! + .. + 1/n!
    static double sum(int n)
    {
        double sum = 0;
        int fact = 1;
        for (int i = 1; i <= n; i++)
        {
            // Update factorial
            fact *= i;

            // Update series sum
            sum += 1.0/fact;
        }
        return sum;
    }

    // Driver program
    public static void main (String[] args)
    {
        int n = 5;
        System.out.println(sum(n));
    }
}

// This code is contributed by Ajit.

Python3

# Python3 program to compute sum of series
# 1/1! + 1/2! + .. + 1/n!

# Function to return value of
# 1/1! + 1/2! + .. + 1/n!
def sum(n):
    sum = 0
    fact = 1

    for i in range(1, n + 1):

        # Update factorial
        fact *= i

        # Update series sum
        sum += 1.0/fact

    print(sum)

# Driver program to test above functions
n = 5
sum(n)

# This code is contributed by Danish Raza

C#

// A simple C# program to compute sum
// of series 1/1! + 1/2! + .. + 1/n!
using System;

class GFG {

    // An Efficient Function to return
    // value of 1/1! + 1/2! + .. + 1/n!
    static double sum(int n)
    {
        double sum = 0;
        int fact = 1;

        for (int i = 1; i <= n; i++)
        {

            // Update factorial
            fact *= i;

            // Update series sum
            sum += 1.0 / fact;
        }
        return sum;
    }

    // Driver program
    public static void Main ()
    {
        int n = 5;

        Console.WriteLine(sum(n));
    }
}

// This code is contributed by Sam007.

PHP

<?php
// A simple PHP program to
// compute sum of series
// 1/1! + 1/2! + .. + 1/n!

// An Efficient Function to
// return value of 1/1! +
// 1/2! + .. + 1/n!
function sum($n)
{
    $sum = 0;
    $fact = 1;
    for ($i = 1; $i <= $n; $i++)
    {
        // Update factorial
        $fact *= $i;

        // Update series sum
        $sum += 1.0 / $fact;
    }
    return $sum;
}

// Driver Code
$n = 5;
echo sum($n);

// This code is contributed by vt_m.
?>

Javascript

<script>

    // A simple Javascript program to compute sum
    // of series 1/1! + 1/2! + .. + 1/n!

    // An Efficient Function to return
    // value of 1/1! + 1/2! + .. + 1/n!
    function sum(n)
    {
        let sum = 0;
        let fact = 1;

        for (let i = 1; i <= n; i++)
        {

            // Update factorial
            fact *= i;

            // Update series sum
            sum += 1.0 / fact;
        }
        return sum.toFixed(5);
    }

    let n = 5;

      document.write(sum(n));

</script>

Output: 

1.71667

Time complexity: O(n) since using a single loop

Auxiliary Space: O(1), since no extra space has been taken.
This article is contributed by Rahul Gupta. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.