Exam 11: Taylor Polynomials and Infinite Series

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

Find the general term of the sequence. ​

Find P​₃(x), the third Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions.

Use the integral 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 five decimal places, if necessary. ​

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 , .

Determine whether the following p-series is convergent or divergent. ​ ​

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

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

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

Write down the first five terms of the sequence. ​

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 P​₄(x), the fourth Taylor polynomial of the following function at the indicated point. ​ at ​ Express any non-integer coefficients as reduced fractions.

The proprietor of Qwik Film Lab recently purchased $12,400 of new film-processing equipment. She expects that this investment, which has a useful life of 4 yr, will yield returns of $4,400 at the end of the first year, $5,200 at the end of the second year, $4,300 at the end of the third year, and $3,100 at the end of the fourth year. Find the internal rate of return on the investment. ​ Round your answer to the nearest hundredth, if necessary. ​ __________ % per yr

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

The temperature at 5 A.M. on a certain December day was measured at 18°F. In the next t hr, the temperature was given by the function ​ ​ where T is measured in degrees Fahrenheit. At what time was the temperature 0 °F? ​ _____ : _____ P.M.

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

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

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