Finding sum of sequence upto a specified accuracy using JavaScript

We need to find the sum of a sequence where each term is the reciprocal of factorials. The sequence is: 1/1, 1/2, 1/6, 1/24, ... where the nth term is 1/n!.

Understanding the Sequence

The sequence can be written as:

• 1st term: 1/1! = 1/1 = 1
• 2nd term: 1/2! = 1/2 = 0.5
• 3rd term: 1/3! = 1/6 ? 0.167
• 4th term: 1/4! = 1/24 ? 0.042

Each term is 1 divided by the factorial of its position number.

Mathematical Formula

Sum = 1/1! + 1/2! + 1/3! + ... + 1/n!

Implementation

const num = 5;

const seriesSum = (n = 1) => {
   let sum = 0;
   let factorial = 1;
   
   for (let i = 1; i <= n; i++) {
      factorial = factorial * i;  // Calculate i!
      sum += 1 / factorial;       // Add 1/i! to sum
   }
   
   return sum;
};

console.log("Sum of first", num, "terms:", seriesSum(num));

// Let's see individual terms
for (let i = 1; i <= num; i++) {
   let fact = 1;
   for (let j = 1; j <= i; j++) {
      fact *= j;
   }
   console.log(`Term ${i}: 1/${fact} = ${(1/fact).toFixed(6)}`);
}

Sum of first 5 terms: 1.7166666666666666
Term 1: 1/1 = 1.000000
Term 2: 1/2 = 0.500000
Term 3: 1/6 = 0.166667
Term 4: 1/24 = 0.041667
Term 5: 1/120 = 0.008333

Optimized Version with Accuracy Control

For better precision control, we can stop when terms become negligibly small:

const seriesSumWithAccuracy = (maxTerms = 10, accuracy = 0.000001) => {
   let sum = 0;
   let factorial = 1;
   let term;
   
   for (let i = 1; i <= maxTerms; i++) {
      factorial = factorial * i;
      term = 1 / factorial;
      sum += term;
      
      console.log(`Term ${i}: ${term.toFixed(8)}, Running sum: ${sum.toFixed(8)}`);
      
      // Stop if term is smaller than required accuracy
      if (term < accuracy) {
         console.log(`Stopped at term ${i} (accuracy reached)`);
         break;
      }
   }
   
   return sum;
};

console.log("\nFinal sum:", seriesSumWithAccuracy(10, 0.0001));

Term 1: 1.00000000, Running sum: 1.00000000
Term 2: 0.50000000, Running sum: 1.50000000
Term 3: 0.16666667, Running sum: 1.66666667
Term 4: 0.04166667, Running sum: 1.70833333
Term 5: 0.00833333, Running sum: 1.71666667
Term 6: 0.00138889, Running sum: 1.71805556
Term 7: 0.00019841, Running sum: 1.71825397
Stopped at term 7 (accuracy reached)

Final sum: 1.7182539682539683

Mathematical Significance

This series converges to e - 1 ? 1.71828, where e is Euler's number (? 2.71828). The series 1 + 1/1! + 1/2! + 1/3! + ... equals e.

Conclusion

The sum of reciprocals of factorials converges quickly to e - 1. Using accuracy control helps determine when sufficient precision is achieved without unnecessary computation.