Exam 11: Taylor Polynomials and Infinite Series

Find the general term of the sequence. ​

Determine whether the following series is convergent or divergent. ​

Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

Use the comparison test to determine whether the following series is convergent or divergent. ​

Write down the first five terms of the sequence. ​

Consider the function at . ​ Use the second Taylor polynomial to approximate . Round the answer to three decimal places, if necessary. ​ __________ ​ Find a bound for the error in the approximation. Round the answer to five decimal places, if necessary. ​ __________

Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

Use the comparison test to determine whether the following series is convergent or divergent. ​ ​

Consider the series ​ . ​ Determine whether the geometric series converges or diverges. ​ If it converges, find its sum.

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary. ​

Suppose the average wage earner saves 15% of her take-home pay and spends the other 85%. Estimate the impact that a proposed $25 billion tax cut will have on the economy over the long run due to the additional spending generated. Round the answer to the nearest billion dollars. ​ $ __________ billion

Consider the function at . Round answers to eight decimal places, if necessary. ​ Use the second Taylor polynomial to approximate . ​ __________ ​ Find a bound for the error in the approximation. ​ __________

Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​

Find the Taylor series of the function at the indicated point. Give the interval of convergence for the series. ​

Find P​₂(x), the second Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions.

Use the comparison test to determine whether the following series is convergent or divergent. ​

Estimate the value of the radical by using three iterations of the Newton-Raphson method with the indicated initial guess for the function. Round your answer to six decimal places, if necessary. ​ ​

Consider the series . ​ Find the radius of convergence of the power series. ​ __________ ​ Find the interval of convergence of the power series. ​ __________

Use the Newton method to approximate the indicated zero of the function. Continue with the iteration until two successive approximations differ by less than 0.0001. Round your answer to five decimal places, if necessary. ​ The zero of between and , .