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

Explain your reasoning!

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

with center x_{0}=3.

Problem 3 (15 points)
Let . Find the Taylor series of f(x) with center x_{0}=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 Q_{2}, Q_{3}, Q_{4}, where Q_{n} is the amount (in mg) of the antibiotic in the body after the
tablet is taken. Note that Q_{1}=250 mg.

2.

Write an expression for Q_{n}, 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.

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