Sum of Series 1/1! + 2/2! + ... + N/N! in C: While & Do-While Loops Explained

In this tutorial, you will learn how to find the sum of the series 1/1! + 2/2! + ... + N/N! in C programming. We will use while and do-while loops to solve this. It is a great way to practice how to use loops inside other loops (nested loops).

To solve this, we need to calculate the factorial for each number. If you have tried this before and your answer kept coming out as "0", don't worry! This happens because of how C handles division. We will fix that together.

How the logic works

We use two loops to solve this step-by-step:

• Outer Loop:

  This counts from 1 up to N. Each time it runs, it picks the next number for our series.

• Inner Loop:

  This does the hard work of calculating the factorial (like 1!, 2!, 3!) for the current number.

• The "Float" Fix:

  In C, if you divide two whole numbers, it throws away the decimal. We change the number to a float so we get an accurate answer instead of a zero.

Steps to solve this in C

We simply go through each number, find its factorial using our inner loop, divide the number by its factorial, and add that result to our total sum.

💻 C Program: Sum of Series

#include <stdio.h>

int main() {
    int i = 1, j, n, fact;
    float total = 0.0;

    printf("Enter the value of N: ");
    scanf("%d", &n);

    while(i <= n) {
        fact = 1;
        j = 1;
        
        // Inner loop calculates factorial for the current term
        while(j <= i) {
            fact = fact * j;
            j++;
        }
        
        // Use (float) to avoid integer division errors!
        total = total + ((float)i / fact);
        
        printf("%d/%d!", i, fact);
        if(i < n) printf(" + ");
        i++;
    }

    printf("\nSum of Series = %.2f\n", total);
    return 0;
}
        

📌 Expected Output

Enter the value of N: 3
1/1! + 2/2! + 3/6!
Sum of Series = 2.50
    

🧠 How the Code Works

• The Outer Loop:

  Counts from 1 up to N. Each time it loops, it handles one part of the series, like 1/1! or 2/2!.

• Resetting Factorial:

  Notice how we set fact = 1 inside the outer loop. This is very important—if you don't reset it, your math will be wrong for the next part of the series.

• The "Float" Secret:

  This is the trickiest part! By writing (float)i, we tell C to handle the math as a decimal. If we don't do this, C will treat it as a whole number and round your answer down to zero.

🚀 Ready for a Challenge?

Now that you have the program working, try these small changes to test your skills:

• Modify:

  Can you change the loop to start from 0 instead of 1?

• Format:

  Can you update the printf to show your final answer rounded to 4 decimal places?

• Switch:

  Try rewriting this same logic using a for loop instead of while to see how it looks.

Frequently Asked Questions

Q: How do I calculate the sum of the series 1/1! + 2/2! + ... + N/N! in C?
A: Use a nested loop where the outer loop manages the series terms and the inner loop calculates the factorial. Remember to use (float) casting to ensure accuracy.

Q: Why is my factorial series sum showing 0?
A: This happens due to integer division. You must cast the numerator to a float (e.g., (float)i / fact) so C doesn't discard the decimal part.

Q: Should I use a for loop or a while loop for this?
A: Both work well. A 'for' loop is often cleaner for series problems, but a 'while' loop is excellent for beginners to practice step-by-step logic.