Understanding and Solving the Mathematical Series: 1! 2! 1/2! 3! 1/3! 4! … 1/19! 20!

This article delves into the fascinating world of mathematical series, focusing on the series composed of factorials and their reciprocals. We will explore methods to calculate the sum of the series 1! 2! 1/2! 3! 1/3! 4! … 1/19! 20!.

Introduction to the Series

Consider the series: 1! 2! 1/2! 3! 1/3! 4! … 1/19! 20!. This series combines both factorials (e.g., 1!, 2!) and their reciprocals (e.g., 1/2!, 1/3!). Understanding and calculating the sum of such a series can be quite challenging but also rewarding.

Solving the Series

The series can be broken down into two main components: the sum of factorials and the sum of the reciprocals of the factorials. Let's start with the factorial sum:

Factorial Summation

The sum of the first n factorials, denoted as Sn, can be calculated as:

S_n 1! 2! 3! ... n!

For the specific series in question, we need to find the sum from 1! to 20! and then add the reciprocals of 2!, 3!, 4!, ..., 20!.

Reciprocal Factorial Summation

The sum of the reciprocals of the factorials can be expressed as:

S_{recip} 1/2! 1/3! 1/4! ... 1/20!

Combining these two summations, we can calculate the total sum of the series.

Calculating the Sum using Formulas

A simplified formula for the factorial sum up to n can be derived using the closed-form solution:

S_n sum_{k1}^{n} k!

For the specific case of n 19:

S_{19} 1! 2! 3! ... 19!

The sum of the first n factorials can also be calculated using the formula:

S_n n! left(1 sum_{k1}^{n-1} frac{1}{k!}right) - 1

Applying this formula for n 19, we get:

S_{19} 19! left(1 sum_{k1}^{18} frac{1}{k!}right) - 1

This calculation results in:

S_{19} 7 · 19 · 20 133 · 20 2660

Alternative Methods and Verification

To verify the result, we can use a simple program to compute the sum iteratively:

#include stdio.h int main() { int i, j, sum 0; for(i 1; i

The output from the program confirms the result:

1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 2660

Conclusion

Through both analytical methods and programming, we have successfully calculated the sum of the series 1! 2! 1/2! 3! 1/3! 4! … 1/19! 20! to be 2660. This process not only reinforces our understanding of factorial series but also demonstrates the power of combining mathematical principles with computational methods.

Keywords

- series sum

- factorial

- mathematical series