19 Practice Exam: Series and Taylor Series
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!

Solution: Taking derivatives, if necessary, we obtain that the fourth degree Taylor polynomial for equals . Using the substitution y=2x, the Taylor polynomial for becomes

Consequently for ,

Thus

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

with center x0=3.

Solution: We have to find the first three derivatives of f(x) at x0=3. , so . , so . , so . Finally, , so .

Thus the third degree Taylor polynomial with center x0=3 is given by

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

Solution: This is easiest if you remember that the Taylor series with center y0=0 for has radius of convergence 1 and is given by

Using the substitution y=x2, one then obtains the Taylor series for f(x):

Since , the Taylor series for f(x) will also have 1 as its radius of convergence.

Alternatively, observe that , then write down a geometric series expression for and integrate.

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

Solution: The absolute value of the general term of the series is

Consequently

The series will converge (diverge), if this quantity is less than 1 (bigger than 1).

so the radius of convergence is .

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

Solution:

You can only do this problem if you recognize the given series as a special case (x=1/2) of the Taylor expansion .

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.

Solution: Using an exponential decay model of the form , where t is measured in hours, and the given information that , we can compute . Consequently

In a similar vein, we obtain

and

In general,

In particular,

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Helmut Knaust

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