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]