Approximating definite
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- (a) Evaluate the integral: Hint: = Your answer should be in the form kn, where k is an integer. What is the value of k? d dx —arctan(r) a₁ = a2 = 2 16 x² + 4 · 6²³ a3 = (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= a4 = dr 1 I²+1 (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 12 terms? (Use the alternating series estimation.)arrow_forwardCan you show me how to solve this?arrow_forwardPlease solution speedarrow_forward
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- Typewritten for upvote. Thank youarrow_forward(a) Find a power series for the function f : (0, 0) → R given by f(x) = $in² about the point x = A. Hint: The Taylor series for xH sin x may be helpful. 2. = B. (b) Find the Taylor series for the function f : (0, 00) → R given by f(x) = log x about the point x =arrow_forward(a) Evaluate the integral: k Your answer should be in the form kä, where k is an integer. What is the value of k? 1 Hint: -arctan(2) d dx x² + 1 = a1 (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function 32 f(x) Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. x² + 4 a2 What are the first few terms of S? ao = = || az = 2 32 x² + 4 = = a4 = dx (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 7 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 7 terms? (Use the alternating series estimation.)arrow_forward
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