For each of the following functions, find the Taylor Series about the indicated center and also determine the interval of convergence for the series. 1. f(x) = ex-2, c = 2 Π == 2. f(x) = sin(x), c = 2
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- The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is given by A(t)=a(e)rt, where a is the amount ofprincipal initially deposited into an account thatcompounds continuously. Prove that the percentageof interest earned to principal at any time t can becalculated with the formula I(t)=ert1.Observe the function X f(x) = (1+2x)² In order to find the power series for this function, complete the following steps: 1 1-x a. Start with the series Σ. Replace x with (−2x) in this series and k=0 write the corresponding power series for = 1 1+2x b. Take derivative of the series from part (a) above and relate it to the power series for the function 1 (1+2x)²· c. Multiply both sides of the resulting series from above with x, and obtain the series for Write the first four non-zero terms of this series. X (1+2x)² d. What is the radius of convergence for this series? What is the interval of convergence?Consider the function f(x) = 2 tan ¯¹(x). a. Differentiate the Taylor series about 0 for f(x). b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative.
- Let f(x) = 1 + x 1 X Find the power series representation for the function f(x) by completing the following steps: a. First, express the fraction 1¹ as a power series. = X b. Now, express the fraction as a power series. 1-x 1+x x c. The function f(x) 1-x 1-x + 1 is the sum of the two series from parts (a) and (b). Express the function f(x) as a power series. d. What is the interval of convergence and the radius of convergence for this power series?Find the 3rd degree Maclaurin Polynomial for f(x) = 2". d Recall that dx In(b) - b². T3(x) Then the Maclaurin series for f(x) = 2" is: T(x) = n=0 Your answer may disappear. The bug has been reported and is being worked on. And this series converges on the interval: enter answer in interval notation. Use oo for oo.Give the first four nonzero terms of the series about x = 27 representing the function f ( x ) = 3 √ x Give the first four nonzero terms of the series about x = 0 representing the function f ( x ) = e2 x cos( 3 x )
- Consider the function below. a. Differentiate the Taylor series about 0 for f(x). b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative.Consider the function below. a. Differentiate the Taylor series about 0 for f(x). b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative.Find the series' interval of convergence and, within its interval, the sum of the series as a function of ?x.
- If a power series for some function y = g(x) has interval of convergence (−2, 2], is itpossible for the series for f′(x) to have interval of convergence [−2, 2]?Explain your answer.I for the function: f(x) = cos(7x). = COS a. Use sigma notation to write the Taylor series T(x) about xo = 14 (-1)"+172n+1 Σ fact (2n+1) 2n+1 A T(x) X - 18 n=0 b. Find interval of convergence of the series you found in Part a. Interval of convergence: (-infinity, infinity)Find the power series for each function and identify its interval of convergence. f(x) = 1 – x2 1.