Which Asymptotic Equation is More Accurate for Large Values of Rho?

[EngWiPy]

Hello all,

Which of the following asymptotic equations (as rho goes to infinity) correct:

$$1-\prod_{m=1}^M(1-\rho^{-x_m})\doteq \sum_{m=1}^M \rho^{-x_m}$$

or

$$1-\prod_{m=1}^M(1-\rho^{-x_m})\doteq \rho^{-\underset{m}{\min}x_m}$$

Thanks

 

[mathman]

The first expression is more accurate. The second will do as long as the min x_m appears only once. If it appears more than once, you need to multiply by the number of occurrences.