htg said:Homework Statement
What is n!/(1*3*...*(2*n+1)) for large n ? (I need an expression E(n) such that the ratio of E(n) by the quantity (n!/(1*3*...*(2*n+1)) goes to 1 as n goes to ∞.
Homework Equations
Does anybody know of an analog of Stirling's formula for ∏(2*k+1)?
The Attempt at a Solution
Ray Vickson said:Use the fact that 1*3*5* ... *(2n+1) = (2n+1)!/[2*4*6* ... *2n], and re-write the denominator here.
RGV
For large values of n, the expression n/(1*3*(2*n+1)) approaches 1/6. This is known as the limit of the sequence and can be used to approximate the behavior of the function for large values of n.
As n increases, the value of n/(1*3*(2*n+1)) approaches 1/6, which means the function approaches a horizontal asymptote at y=1/6. This means that the function approaches a constant value as n gets larger.
No, the expression cannot be simplified for large values of n. However, as mentioned earlier, it can be approximated by 1/6 which can make it easier to analyze the behavior of the function for large n.
The purpose of using large values of n is to study the behavior of the function for extreme values. By analyzing the function for large n, we can understand how it behaves when the input is significantly larger than the values we typically encounter.
The expression n/(1*3*(2*n+1)) can be used to approximate the behavior of various systems in engineering, physics, and economics. For example, it can be used to analyze the growth rate of a population over time or the decay of a radioactive substance. It can also be used to model the behavior of financial investments or the efficiency of a chemical reaction.