January 14, 2000
Homework

1.

Determine whether the following series converge or diverge using the limit comparison test:

(a)

$$\sum_{n=0}^{\infty} \frac{n^2 - 2n + 5}{n^4 + 3n^2 + 1}$$

(b)

$$\sum_{n=0}^{\infty} \frac{n+5}{n^2 + 4}$$

2.

Find the sum of the following series:

(a)

$$\sum_{n=0}^{\infty} \frac{1}{(n+3)(n+4)}$$

(b)

$$\sum_{n=0}^{\infty} \frac{1}{(n+1)(n+3)}$$

3.

Find the Taylor series expansion for each of the following functions about the specified point (if you find it helpful, you may use the important examples I gave you in class, rather than deriving them from scratch). Graph the functions, and compute the radius of convergence of the series (you do not have to test the end points - today!). You should be able to find an expression for the general term a_n, but if you do not recognize a pattern, then just write the first six terms of the Taylor series.

(a)

f(x) = e^-3x about x₀ = 0.

(b)

$f(x) = \cos{(\pi x)}$ about x₀ = 0.

(c)

$f(x) = \frac{1}{1+x}$ about x₀ = 0.

(d)

$f(x) = \frac{1}{(2-x)^2}$ about x₀ = 0.

(e)

$f(x) = \sqrt{4 - x}$ about x₀ = 0.

(f)

$f(x) = \ln{(3x - 2)}$ about x₀ = 1.

(g)

$f(x) = \cos{2x}$ about $x_0 = -\frac{\pi}{6}$