Practice Exam: Series and Taylor Series
Time: 60 minutes

Problem 1 (15 points)   Use the fourth degree Taylor polynomial of to find the exact value of

Problem 2 (15 points)   Find the third degree Taylor polynomial of

with center x0=3.

Problem 3 (15 points)   Let . Find the Taylor series of f(x) with center x0=0 and its radius of convergence.

Problem 4 (15 points)   Find the radius of convergence of the power series

Problem 5 (20 points)   Find the exact value of the following series:

Problem 6 (20 points)   An antibiotic decays exponentially in the human body with a half-life of about 2.5 hours. Suppose a patient takes a 250 mg tablet of the antibiotic every 6 hours.

1.
Write an expression for Q2, Q3, Q4, where Qn is the amount (in mg) of the antibiotic in the body after the tablet is taken. Note that Q1=250 mg.
2.
Write an expression for Qn, and put it in closed form.
3.
Assume the antibiotic treatment consists of a total of 28 tablets. Give a numerical estimate for the amount of antibiotic in the body immediately after the patient takes the last tablet of the treatment.