onsider 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.

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=

3. elemfe Consider the function f(x) = 2. (a) Find the Taylor series of f(x) at a = 5 using the definition of a Taylor series. (b) Find the radius of convergence of this power series. 1

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.