MAT136H1 Lecture Notes - Alternating Series Test

(1 point) (a) Evaluate the integral 2 40 dx. Jo x2+.4 Your answer should be in the form kn, where k is an integer. What is the value of k? (Hint: darctan(x)- ar x+1 (b) Now, lets evaluate the same integral using power series. First, find the power series for the functionf(x) = x404 . Then, integrate it from 0 to 2, and call it S. S should be an infinite series. What are the first few terms of S? (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a), you have found an estimate for the value of π in terms of an infinite series. Approximate the value of π by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 8 terms? (Use the alternating series estimation.)

r2n (2n)T and that I. We know the Maclaurin series expansion for cos(x) is given by cos(x) = it converges Vx E R. (a) Use the Maclaurin series for cos(x) to express / x cos(z3Jdz as an infinite series. 0b) Uie the Altermating Sris Esiatin Tharund af the ernr ia using your series to estimate the value of the definite integral if you truncate it after N terms. (You may assume that the conditions of the Alternating Series Test are satisfied.) 2. Find the Taylor series, centred about the indicated value, for each of the following functions (you may manipulate known Taylor/Maclaurin series or directly use the definition): (a) f(x) = e, centred at x = 3. (b) f(x) = cos(x), centred at x = 2.

Spring18 Section 10.3: Problem 11 PreviousS Problem List Next (1 point) The goal of this problem is to approximate tcos(t2) dt, and determine the amount of error in our approximation. (a) Use the first two nonzero terms of the taylor polynomial for t2 cos(t2) dt to approximate its value 0 t" cos 0 (b) Determine how close your answer to part (a) is to t2cos(t2) dt using the error bounds for alternating series. 0 The error is at most