Finding the Sum of Series (e.g., 1/1! + 2/2! + …)

Steps to Understand the Program:

1. Factorial: The factorial of a number nnn is the product of all positive integers less than or equal to nnn. For example, 3!=3×2×1=63! = 3 \times 2 \times 1 = 63!=3×2×1=6.
2. Series Expansion: The given series is:S=11!+22!+33!+⋯+nn!S = \frac{1}{1!} + \frac{2}{2!} + \frac{3}{3!} + \cdots + \frac{n}{n!}S=1!1​+2!2​+3!3​+⋯+n!n​where each term kk!\frac{k}{k!}k!k​ involves calculating both the numerator kkk and the factorial of kkk.
3. Logic of Program:
   • We will use a loop to calculate each term in the series.
   • For each kkk, calculate the factorial k!k!k!, then compute kk!\frac{k}{k!}k!k​, and add this value to the sum.

#include <stdio.h>

// Function to calculate the factorial of a number
long double factorial(int n) 
{
    long double fact = 1;
    for (int i = 1; i <= n; i++) 
{
        fact *= i;
    }
    return fact;
}

int main() 
{
    int n;
    long double sum = 0.0;

    // Taking input from the user
    printf("Enter the value of n: ");
    scanf("%d", &n);

    // Loop to calculate the sum of the series
    for (int i = 1; i <= n; i++) {
        sum += (i / factorial(i)); // Calculating each term of the series
    }

    // Display the result
    printf("Sum of the series is: %.10Lf\n", sum);
    return 0;
}

Explanation of the Program:

1. Factorial Function:
   • The function factorial(int n) computes the factorial of a given number nnn by multiplying all integers from 1 to nnn. This result is returned as the factorial of nnn.
2. Main Function:
   • The program starts by prompting the user to enter the value of nnn (the number of terms in the series).
   • A for loop is used to iterate from 1 to nnn, calculating each term of the series.
   • For each iteration iii, the program calculates ii!\frac{i}{i!}i!i​ using the factorial function and adds this value to the variable sum.
   • After the loop completes, the final sum of the series is printed.
3. Data Types:
   • The data type long double is used for storing the sum and factorial values because factorials grow very quickly, and using a regular float or double may not be precise enough for larger numbers.
4. Precision:
   • In the output, the sum is printed with a precision of 10 decimal places (%.10Lf), ensuring that even small values in the series are captured accurately.

Example Output:

Enter the value of n: 5
Sum of the series is: 2.7083333333

Here, for n=5n = 5n=5, the series becomes:S=11!+22!+33!+44!+55!=1+1+0.5+0.1667+0.0417=2.7083S = \frac{1}{1!} + \frac{2}{2!} + \frac{3}{3!} + \frac{4}{4!} + \frac{5}{5!} = 1 + 1 + 0.5 + 0.1667 + 0.0417 = 2.7083S=1!1​+2!2​+3!3​+4!4​+5!5​=1+1+0.5+0.1667+0.0417=2.7083

Explanation of Output:

• The program computes the sum of the first 5 terms of the series and returns the result. The result is the sum of the fractions computed from each term of the series.

This C program efficiently calculates the sum of the series by iterating through each term and using a function to calculate factorials.