(20 pts) By completing this problem, you will have a better understanding why complex analysis is a suitable environment for studying series. (a) Consider the function f ( x ) = 1 1 + x 2 defined in R. Find its Taylor series in real-sense, i.e., as you learned in Calculus. (b) Notice that f(x) defined above is differentiable everywhere in R, but does its Taylor series converge pointwise everywhere in R? (c) What is the Taylor series expansion for f(z) = 1 1 + z 2 in D = {z ∈ C : |z| < 1}, and indicate what is its radius of convergence? (d) What is the Laurent series expansion for f(z) = 1 1 + z 2 in D = {z ∈ C : |z| > 1} ? (e) [no points] Can you explain what happens in (b)?
According to a survey, the heights of adult males in a certain town are normally distributed with mean 170 centimetres and standard deviation 4 centimetres. What is the probability that an adult male chosen at random is between 165 cm and 176 cm tall? What percentage of the adult male population is more than 180 cm tall?
According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. (a) What is the probability that an adult male chosen at random is between 64 and 74 inches tall? (Round your answer to three decimal places.) (b) What percentage of the adult male population is more than 6 feet tall? (Round your answer to one decimal place.) %
According to a survey, the mean height for men is 68.6 inches. In a sample of 320 men between the ages of 60 and 69 , the mean height was x¯ = 68.5 inches. Public health officials want to determine whether the mean height μ for older men is less than the mean height of all adult men. Assuming the population standard deviation to be σ = 2.85 inches, compute the value of the test statistic. 429.95 −0.63 0.63 0.74
According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. (a) What is the probability that an adult male chosen at random is between 66 and 72 inches tall? (Round your answer to three decimal places.) (b) What percentage of the adult male population is more than 6 feet tall? (Round your answer to one decimal place.) %