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Find all the values of x such that the given series would converge. 00 9331?, 2() 11:1 The series is convergent from m = [j , left end included (enter Y or N): to a: = C] , right end included (enter Y or N): Find all the values of x such that the given series would converge. :3 ' \"2\" \" n:1('\\/ + 5) The series is convergent from m = C] , left end included (enter Y or N): to a: = C] , right end included (enter Y or N): Find all the values of x such that the given series would converge. 5: (w - 5\"\" nl 5n The series is convergent from m = C] , left end included (enter Y or N): to a: = C] , right end included (enter Y or N): I Find the interval of convergence for the given power series. i (a: 11)\" 71,21 \"(7)\" The series is convergent from :1: = C] , left end included (enter Y or N): I to a: = C] , right end included (enter Y or N): 7 00 Represent the function as a power series f($) = Z cnm\" 1 2.1: \":0 Compute the first few coefficients of this power series: Find the radius of convergence R = \f40 do: .132 2 (a) Evaluate the integral: / 0 + pk Your answer should be in the form kw, where k is an integer. What is the value of k? Hint: Emctanh = $2+1 k: (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function 40 . Then, integrate it from 0 to 2, and call the result 5. 5 should be an infinite series. What are the first few terms of S? 9 M H '10 (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 7r in terms of an infinite series. Approximate the value of 7r 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.) We are interested in the first few Taylor Polynomials for the function f(a:) : 76m + 46'\" centered at a = 0. To assist in the calculation of the Taylor linear function, T1(:c), and the Taylor quadratic function, T2(a:), we need the following values: m) = me) = m0) = Using this information, and modeling after the example in the text, what is the Taylor polynomial of degree one- Ta What is the Taylor polynomial of degree two: u The Taylor series for f (ac) = ac at -3 is Cn (2 + 3 ) ". n=0 Find the first few coefficients. CO C1 = C2 = C3 = C4Cn(x - 3)n The Taylor series for f(x) = e at a = 3 is n=0 Find the first few coefficients. Co C1 C2 C3 C4The Maclaurin series of the function x) = 43326613 00 can be written as at) : 2 Cum\" n=0 where the first few coefficients are: The Maclaurin series of the function f(x) = 9x2 sin(5ac) can be written as f(a ) = Can n=0 where a few of the coefficients are: C3 C4 = C5 C6 = C7 =The Maclaurin series of the function f(m) = 6 cos(10m2) 00 can be written as at) : Z cnm\" 11:0 where a few of the coefficients are: The Maclaurin series of the function at) 2 5m tan1(9m2) 00 can be written as x) 2 2 Cum\" n20 where a few of the coefficients are: \fSuppose g is a function which has continuous derivatives, and that g(0) = 2, g'(0) = -7, g"(0) = -7 and g"'(0) = -9. What is the Taylor polnomial of degree 2 for g, centered at a = 0? T2 (20 ) = What is the Taylor polnomial of degree 3 for g, centered at a = 0? T3 (20 ) = Use T2 (a) to approximate g(0.2) ~ Use T3 (a) to approximate g(0.2) ~00 The Taylor series for f(a:) = 1n(sec(:c)) at a = O is Z c433)\". 1120 Find the first few coefficients. CF CF Q 622 C4: Find the exact error in approximating ln(sec(0.4)) by its fourth degree Taylor polynomial at a = 0. Find the Taylor polynomial of degree 3, centered at a = 4 for the function x) = V93 1. Let T4(:1:): be the Taylor polynomial of degree 4 of the function f(3) = 111(1 + 3:) at a = 0. Suppose you approximate at) by T4012), find all positive values of x for which this approximation is within 0.001 of the right answer. (Hint: use the alternating series approximation.) in .u ..u l.I.... Find T5(m): Taylor polynomial of degree 5 of the function f(:1:) : cos(:c) at a = O. T500) = Using the Taylor Remainder Theorem, find all values of x for which this approximation is within 0.004414 of the right answer. Assume for simplicity that we limit ourselves to |:r:| S 1. WE Assume that sin(:L-) equals its Maclaurin series for all x. Use the Maclaurin series for sin (71:2) to evaluate the integral 0.73 / sin(7:v2) da: 0 Your answer will be an infinite series. Use the first two terms to estimate its value. :1 Let F(x) = sin (7t2) dt. Find the MacLaurin polynomial of degree 7 for F(x). 0.77 Use this polynomial to estimate the value of sin (7x2) dx. 0