(For those whove thought about convergence issues) Check that the power series expansion f (x) = _ k=0 xk k! converges for any real number x and that f (x) = ex , as follows. a. Fix x _= 0 and choose an integer K so that K 2|x|. Then show that for k > K, we have |x|k k! C _ 12 _ kK, where C = |x| K . . . |x| 2 |x| 1 is a fixed constant. b. Conclude that the series k=K+1 |x|k k! is bounded by the convergent geometric series C j=1 1 2j and therefore converges and, thus, that the entire original series converges absolutely. c. It is a fact that every convergent power series may be differentiated (on its interval of convergence) term by term to obtain the power series of the derivative (see Spivak, Calculus, Chapter 24). Check that f _ (x) = f (x) and deduce that f (x) = ex .

Week 7 Ch. 5 Tying in data with parameters Simple random sample (SRS): a sample of n elements obtained by a method so that any two collections of n elements are equally likely of being selected as the sample - n denotes sample size - you get in big trouble in statistics if you use biased samples - Two types: 1. Sampling with replacement a. Means that...