Chapter 10.1, Problem 72E

Find the radius of convergence of the power series. (-1)"x" En=0| SM

Find the power series for each function and identify its interval of convergence. f(x) = 1 – x2 1.

+(-1)". (X-2)h 2 h+1 Fex) =Ź (x-2) _(x-2) --a. 4 Findea a.values of (x) that make series converged. la. The of series. Note with an explanation of the StePs

If xo is in the interval of convergence of a power series n=1 an (x – c)" that represents a function f(x), which of the following is true? Circle all that apply. (a) The value of f(xo) is approximated by the power series evaluated at x0, and the remain- der is non-zero. (b) The value of f(xo) is exactly equal to the power series evaluated at xo. (c) The interval of convergence must be all real numbers. (d) The only point of convergence is the center of the power series x = c. If f(x) is represented by a power series E-, an (x – c)" over an open interval containing c, then which of the following is true about that power series? Circle all that apply. (a) The remainder of the power series is less than or equal to an. (b) The derivative of the power series is a power series. (c) The interval of convergence must be all real numbers. (d) It must be a Taylor series. (e) The radius of convergence is half the length of the interval of convergence.

Find the interval of convergence for the power series (-1)" Vn 5n (* – 3)". - n=1

Study the convergence of the series

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.

Find the radius and interval of convergence for the power series

a. Find the Taylor series at 0 by performing operations on the basic Taylor series. State the interval of convergence.   ​f(x)=ln(1+16x^2) Type the first four terms and the​ nth-term of the Taylor series.   ​f(x)=   b.Find the Taylor series at 0 by performing operations on the basic Taylor series. State the interval of convergence.   ​f(x)=4/5+x^2 Type the first four terms and the​ nth-term of the Taylor series.   ​f(x)=   c. Evaluating the Taylor series at 0 for f(x)=e^−x at x=0.8 produces the following series.   e^−0.8=1−0.8+0.32−0.085333+0.017067−0.002731+⋯   Use two terms in this series to approximate e^−0.8​, and then estimate the error in this approximation. e^−0.8=