Practice Exam: Series and Taylor Series
Time: 60 minutes

Problem 1 (15 points)   Use the fourth degree Taylor polynomial of $\cos(2x)$ to find the exact value of

$$\lim_{x\to0}\frac{1-\cos(2x)}{3x^2}$$

Explain your reasoning!

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

$$f(x)=\sqrt{x}$$

with center x₀=3.

Problem 3 (15 points)   Let $f(x)=\ln(1+x^2)$. 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

$$\sum_{n=0}^\infty (-3)^n\sqrt{n+1}\,(x+1)^{2n+1}$$

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

• $$\frac{4}{5}-\frac{16}{25}+\frac{64}{125}-\frac{256}{625}+\cdots$$
• $$1+\frac{1}{2}+\frac{1}{4\cdot 2!}+\frac{1}{8\cdot 3!}+\frac{1}{16\cdot 4!}+\frac{1}{32\cdot 5!}+\cdots$$

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 Q_n is the amount (in mg) of the antibiotic in the body after the $n^{\mbox{th}}$ tablet is taken. Note that Q₁=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.

If you would like to check your answers, click on Answer.

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