How I Found the Value of a Product Defined by the Euler-Maclaurin Formula
This article delves into the process of finding the value of a specific product defined through the Euler-Maclaurin formula. We explore the approximation methods and the accuracy of these techniques as the product parameter increases.
Introduction
In analyzing the product f_n(x) prod_{k1}^n left(1 - frac{x^2}{k^2}right), the goal is to determine its behavior as n approaches infinity. This product is significant for understanding the asymptotic behavior of certain mathematical series and functions.
Asymptotic Behavior
It is well-known that as n to infty, the product converges to the function
( frac{sin(pi x)}{pi x} )
and hence, in the case of x frac{1}{2}, the product is approximately
( frac{sin(pi/2)}{pi/2} frac{2}{pi} )
However, the exact behavior of the product for finite values of n is not exactly 2/pi. It is slightly larger, and even more interestingly, the difference from 2/pi increases as n increases.
Analytic Approximations
Using an iterative method related to the Euler-Maclaurin formula, an approximation to f_n(1/2) can be obtained. The first iteration yields:
( log f_n(1/2) frac{2}{pi}n - frac{1}{2}log left(1 - frac{1}{2n^2}right) tanh^{-1} frac{1}{2n} Phi_n )
where ( Phi_n ) decays rapidly to 0 as n to infty. This approximation is already quite accurate for large n.
Numerical Methods and Accuracy
The second iteration is theoretically better but is extremely cumbersome to work out by hand. A computer program or integral algebra could easily handle the second iteration by computing (Phi_n) and (Phi_2) which decay even more quickly.
When n 50, one can achieve accuracy to 5 decimal places by ignoring (Phi_n). This suggests that a more detailed second iteration might be preferable, especially for larger values of n.
Other Approximation Techniques
Multiplying out the product directly could provide a way to achieve higher precision, but it might not be as straightforward as using the Euler-Maclaurin approach.
Conclusion
In summary, finding the value of the product defined by the Euler-Maclaurin formula involves a balance between hand calculations and computational methods. The Euler-Maclaurin approach, though potentially cumbersome, offers a practical solution for large values of n.