Theorem 2.1. For all n ≥ 1, n log n− n < log(n!) < n log n, so log(n!) ∼ n log n. Proof. The inequality log(n!) < n log n is a consequence of the trivial inequality n! < nn. Here are three methods of showing n log n− n < log(n!). Method 1: A Riemann sum approximation for ∫ n 1 log x dx using right endpoints is log 2+ · · ·+ log n = log(n!), which overestimates, so log(n!) > ∫ n 1 log x dx = n l...

obesity usually results from a combination of several modifiable and non-modifiable contributing factors, such as genetics, race, and socioeconomic status. other factors, including birth by cesarean section, perinatal antibiotics usage, and promotion of exclusive breastfeeding for six months, could be potentially prevented or modulated. however, one of the factors can be easily implemented thro...