(a) Evaluate the integral: S Hint: Your answer should be in the form ki, where k is an integer. What is the value of k? d 1 -arctan(r) dx ²+1 k (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. 16 f(x)= x²+4 What are the first few terms of S? ao a1 a2 a3 || || 16 x² + 4 || || dr a4= (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 in terms of an infinite series. Approximate the value of by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 11 terms? (Use the alternating series estimation.)
(a) Evaluate the integral: S Hint: Your answer should be in the form ki, where k is an integer. What is the value of k? d 1 -arctan(r) dx ²+1 k (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. 16 f(x)= x²+4 What are the first few terms of S? ao a1 a2 a3 || || 16 x² + 4 || || dr a4= (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 in terms of an infinite series. Approximate the value of by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 11 terms? (Use the alternating series estimation.)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.1: Infinite Sequences And Summation Notation
Problem 73E
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