Assignment12: Problem 1 (1 point) Consider the series S (7x) Find the interval of convergence of this power series by first using the ratio test to find its radius of convergence and then testing the series' behavior at the endpoints of the interval specified by the radius of convergence. Radius of convergence R = Interval of convergence I = (Enter your answer for I as an interval: thus, if the interval of convergence were -3 e n1 Radius of convergence R =| | Interval of convergence I = | (Enter your answer for I as an interval: thus, if the interval of convergence were 3 | | n0 s =+ J+ [+ J+[J+- The open interval of convergence is: |:| (Give your answer in interval notation. ) Using Interval Notation Using Interval Notation = [fan endpoint is included, then use [ or ]. If not, then use ( or ). For example, the interval from -3 to 7 that includes 7 but not -3 is expressed (-3,7]. For infinite intervals, use Inf for @ (infinity) and/or - Inf for =@ (-Infinity). For example, the infinite interval containing all points greater than or equal to 6 is expressed [6,Inf). If the set includes more than one interval, they are joined using the union symbol U. For example, the set consisting of all points in (-3,7] together with all points in [-8,-5) is expressed [-8,-5)U(-3,7] If the answer is the empty set, you can specify that by using braces with nothing inside: { } You can use R as a shorthand for all real numbers. So. it is equivalent to entering ( - Inf, Inf). = You can use set difference notation. So, for all real numbers except 3. you can use R- {3} or (-Inf, 3)U(3,Inf) (they are the same). Similarly, [1,18)-{3,4} is the same as [1,3)U(3,4)U(4,18). WeBWork will not interpret [2,4]U[3,5] as equivalent to [2,5]. unless a problem tells you otherwise. All sets should be expressed in their simplest interval notation form, with no overlapping intervals. webwork? files/helpFiles/inte rvalMotation. him Assignment12: Problem 8 (1 point) 2 Represent the function (1 - 7x)2 as a power series f(*) = > Can n 0 CO C1 CA Find the radius of convergence R =Assignment12: Problem 9 (1 point) The function f(z) = 10z In(1 + z) is represented as a power series oo flz) = z enT'. n0 Find the following coefficients in the power series. 2=| | es=| | ea=| | = | co=| | Find I_E_radius of convergence R of the series. R=| | Assignment12: Problem 10 (1 point) Use sigma notation to write the Maclaurin series for the function. In(9 + =). Maclaurin series | | + Z | | L) &=L (Note first term separate and summation from k = 1) Assignment12: Problem 11 (1 point) Write out the first four terms of the Maclaurin series of f() if f(0) = -8, f'(0) = 13, f"(0) = -7, f"(0) = 9 f ( I) = . ..Assignment12: Problem 12 (1 point) Match each of the Maclaurin series with the function it represents. 00 n! 7 0 (-1)2 241 2. 2n + 1 n 0 3. S (-1)n2 2n n 0 (2n)! DO 4. [(-1) 12 2 2041 1 0 (2n + 1)! A. cos(x) B. er C. sin(I) D. arctan(x)Assignment12: Problem 13 (1 point) 6+ 7: Let f(x) = Compute f(I) f(1) = f'(x) = f'(1) = f"(I) f" (1) = f" (I) = f" (1) = f (iv) ( x) = f(iv) (1) = f() (x) = f() (1) = We see that the first term does not fit a pattern, but we also see that f() (1) = for k > 1. Hence we see that the Taylor series for f centered at 1 is given by f(x) = 13+ (x - 1)* * 1\fAssignment12: Problem 15 (1 point) Use sigma notation to write the Maclaurin series for the function, ed Maclaurin series =Assignment12: Problem 16 (1 point) D Find the first four terms of the Taylor series for the function about the point @ = 3. (Your I answers should include the variable x when appropriate.) Assignment12: Problem 17 (1 point) The Taylor series for f(x) = In(sec(x) ) at a = 0 is) Find the following coefficients. CO C1 C2 C3 CAAssignment12: Problem 18 (1 point) Represent the function 6 In(3 =) as a power series (Maclaurin series) f(z) = {'1]| | = | 2= | es=| | es=| | Find the radius of convergence R =| | Assignment12: Problem 19 (1 point) Find the Maclaurin series and corresponding interval of convergence of the following function. f(I) = 1 - 5x f(I) = > The interval of convergence is