Exploring the Value of 1-1/3 × 1-1/4 × … 1-1/n: A Comprehensive Guide

Understanding the value of the product series 1 - frac{1}{3} times 1 - frac{1}{4} times ldots times 1 - frac{1}{n} is quite intriguing and can be a great way to explore mathematical patterns and limits. This guide will walk you through the step-by-step solution and provide insights, including how to derive the product and its behavior as n increases.

Solution

Let's start by analyzing the individual terms in the product series:

For any term of the form 1 - frac{1}{k}, it can be simplified as:

1 - frac{1}{k} frac{k-1}{k}

Now, let's see how this applies to the first few terms in the product:

First Few Terms

1 - frac{1}{3} frac{2}{3}

1 - frac{1}{4} frac{3}{4}

1 - frac{1}{5} frac{4}{5}

By multiplying these terms together, we get:

left(frac{2}{3}right) times left(frac{3}{4}right) times left(frac{4}{5}right) times ldots times left(frac{n-1}{n}right)

Pattern Recognition

Notice the pattern in the product:

frac{2 times 3 times 4 times ldots times (n-1)}{3 times 4 times 5 times ldots times n}

Here, the numerator and denominator partially cancel each other out, leaving us with:

frac{2}{n}

Conclusion

Therefore, the product of the first n terms of the series is:

boxed{frac{2}{n}}

The Code Implementation

For those who prefer a practical approach, here is a simple code snippet to simulate the product calculation:

#include stdio.h#define n 100int main() {    double t  2.0 / 3.0;    for (int i  2; i 

Running this code with different values of n gives the following results:

For n 10, the result is approximately 0.133333 For n 100, the result is approximately 0.013333 For n 1000, the result is approximately 0.001333 For n 10000, the result is approximately 0.000133

The larger the value of n, the closer the result approaches 0, which is consistent with the mathematical limit.

Discussion

The behavior of the series can be further understood by analyzing the product frac{2}{n}. As n increases, the value of frac{2}{n} decreases, confirming that the product tends towards 0 as n approaches infinity.

Wrap Up

This example not only helps in understanding mathematical series but also in recognizing patterns and limits. Whether you are a student or a professional, this exploration into the nature of series can be a valuable learning experience.