In this problem it will be useful to recall the following properties of logarithms: log(xy)=log(x)+log(y) and log(xa)=alog(x).
Find the least integer k such that f(n) is O(nk) for each of the following functions:

f(n) = 18 + 28 + � + n8

f(n) = log(n!)

f(n) = log(nn) / (n9 + 1)

Thompson and thompson is a steel bolts manufacturing company. their current steel bolts have a mean diameter of 127 millimeters, and a variance of 36. if a random sample of 35 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by greater than 0.5 millimeters? round your answer to four decimal places.