136.275, Assignment No. 2

October 18, 2002

The assignment is due Friday, October 25 , 2002 in class. Late assignments receive a mark zero.

1. Suppose that the series converges and the series diverges.

a) Show that the series diverges. [5]

b) Find examples to show that if and both diverge, then the series

may either converge or diverge. [3]

2. Use the Integral Test to determine for which values of p does the series

converge.

( You have to show that you can use the Integral Test.) [7]

3. Determine if the following series converge or not:

a) , [3]

b) , [4]

c) [3]

d) . [3]

4. Find the radius of convergence and the interval of convergence for the series

. [6]

5. a) Find the power expansion of about . Show that it converges to for

all x in |R by using the Remainder Theorem. [6]

b) Can we expand 1/x about ? Can the function 1/x be represented by a power

series about on an interval with radius greater than 1? Explain. [4]

Total [ 44/42]