x(1-1%2F3%5E2)x(1-1%2F4%5E2)-to-infinity.jpeg "(1-1/2^2)x(1-1/3^2)x(1-1/4^2) To Infinity")
An Infinite Product: Unveiling the Mystery of (1-1/2^2)x(1-1/3^2)x(1-1/4^2)...
At first glance, the infinite product (1-1/2^2)x(1-1/3^2)x(1-1/4^2)... might seem daunting. How can we possibly find a finite value for a product that goes on forever? The answer lies in a clever observation and some basic algebraic manipulations.
The Key Observation
Notice that each term in the product can be factored:
- (1 - 1/2^2) = (1 + 1/2)(1 - 1/2) = (3/2) * (1/2)
- (1 - 1/3^2) = (1 + 1/3)(1 - 1/3) = (4/3) * (2/3)
- (1 - 1/4^2) = (1 + 1/4)(1 - 1/4) = (5/4) * (3/4)
And so on...
A Pattern Emerges
When we multiply these factors together, we see a beautiful cancellation pattern:
- (3/2) * (1/2) * (4/3) * (2/3) * (5/4) * (3/4) ...
Notice that the '2' in the denominator of the first term cancels with the '2' in the numerator of the second term, the '3' in the denominator of the second term cancels with the '3' in the numerator of the third term, and so on.
The Result
This cancellation leaves us with only the first term in the numerator (3/2) and a sequence of terms in the denominator that approaches infinity.
Therefore, the infinite product (1-1/2^2)x(1-1/3^2)x(1-1/4^2)... equals 3/2.
Conclusion
This fascinating example demonstrates the power of recognizing patterns and manipulating expressions. By factoring and observing cancellations, we were able to evaluate an infinite product and arrive at a surprising, finite result. It's a testament to the beauty and elegance of mathematics!