Math 3112
Test 3
Fall 1996

No books, notes etc. are permitted.
Show all your work! Box in your answers!
The test has 6 problems on 4 pages.
Read the problems very carefully.

Problem 1 (15 points) Find the second-degree Taylor polynomial with center x₀=0 for the function $f(x)=\sec(x)$.

Problem 2 (15 points) Compute the sum of the following series. (Give an exact answer, not a numerical approximation!)

$$\frac{3}{8}+\frac{9}{32}+\frac{27}{144}+\frac{81}{576}+\cdots$$

Oops! Replace 144 by 128 and 576 by 512.

Problem 3 (20 points) (A) Find the Taylor series with center x₀=0 of the function

$$f(x)=\frac{3}{2+x}.$$

(B) What is the radius of convergence of the series in (A)?

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

$$\sum_{n=1}^{\infty}\frac{(-1)^n}{n \cdot 3^{n+1}}(x-1)^{2n} .$$

Problem 5 (20 points) Find the Taylor series with center x₀=0 of the following functions. You should be able to predict the general term of the Taylor series.

(A) $$f(x)=\arctan(2x)$$

(B) $$f(x)=\frac{1}{(1-x)^2}$$

(C) $$f(x)=x\cdot \ln(1+x)$$

Problem 6 (15 points) Compute (all terms of) the Fourier series of the f(x)=|x| on the interval $[-\pi,\pi]$.
Hint: Note that for $k=1,2,3,\ldots$ the function $\vert x\vert \sin(kx)$ is odd, while $\vert x\vert \cos(kx)$ is even!

Extra-Credit Problem (15 points) Show that the derivative of any even function is odd, while the derivative of any odd function is even. Use this to show that any even function's Taylor series with center 0 consists only of ``even terms".