SCIE 001 MATHEMATICS ASSIGNMENT 13 (Due Mar. 25, 2013 ) There are two parts to this assignment. The first part is on WeBWorK and is due by 10:00 am on Mon. Mar. 25. The second part consists of the questions on this page, and is due before 10:00 am on Mon. Mar. 25. For these questions, you are expected to provide full solutions with complete arguments and justifications. You will be graded primarily on the correctness, clarity and elegance of your solutions. Your answers must be typeset or very neatly written. They must be stapled, with your name and student number at the top of each page. In this assignment you may use the Alternating Series Test (p. 727) and the Alternating Series Estimation Theorem (p. 730) which can be combined as follows: If 0 < bn+1 ≤ bn for all positive integers n and limn→∞ bn = 0, then the sum of the alternating series s= ∞ X n=1 (−1)n−1 bn = b1 − b2 + b3 − b4 + . . . exists,Pand the remainder Rn = s − sn or error made by approximating the sum s by the nth partial sum sn = ni=1 (−1)i−1 bi satisfies |Rn | ≤ bn+1 . (The proof of the first result is much like question 3 of Assignment 12, and the second result is basically the observation that s must be between any two consecutive partial sums, since the partial sums “bounce” above and below the final limit with decreasing size of bounces.) R 1. (a) Find Maclaurin series for 1/(1 + x2 ) and arctan(x) (noting that 1/(1 + x2 ) dx = arctan(x) + C), and their radii of convergence. (b) Does the Maclaurin series for 1/(1 + x2 ) converge at x = 1? Does the Maclaurin series for arctan(x) converge at x = 1? Explain. (c) Show that ∞ X 4 4 4 4 (−1)n−1 . π = 4− + − +··· = 3 5 7 2n −1 n=1 2. In probability, a random variable is said to be normally distributed with mean µ and standard deviation σ (where µ is any real number and σ > 0) if its probability density function is the function (p. 572) f (x) = 2 2 1 √ e−(x−µ) /(2σ ) . σ 2π (Measurements made in labs are often assumed to be normally distributed random variables.) (a) The probability that a normally distributed random variable is “within one standard deviation of the mean” is Z µ+σ P = f (x) dx. µ−σ Make a substitution z = (x − µ)/σ to get 1 P =√ 2π 2 Z 1 e−z 2 /2 dz. −1 (b) Use the Maclaurin series for e−z /2 to find P accurate to within 0.005 of its exact value (without knowing what the exact value of P is). 3. (Bonus) Explain what is wrong with the textbook’s proof of the Alternating Series Test on p. 728 and suggest how it could be fixed.