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MA 126 NAME_________________________________
Closed Book, No Calculators Fall, 2008
PART 1. Part 1 consists of 6 questions. Do your work and clearly write your answer in the space
provided. No partial credit is awarded for this part of the test. (5 points each)
- Determine whether the geometric series 1
5 n
∞
∑ is convergent or divergent. If it is convergent, find
its sum.
Answer: _____________________
- Determine whether the series k 1 2 1
n n
∞
=
∑ is convergent or divergent. If it is convergent find its sum.
Answer: _____________________
- Determine whether the sequence given by n ln
n a n
= converges or diverges. If it is convergent, find
its limit. Answer:________________________
- Find the radius of convergence of the series 0!
n
n
x n
∞
=
Answer:__________________________
- If the series (^) 2 1
( 1) n n n
∞
=
∑ were approximated by its third partial sum, what would be the maximum error
to expect?
Answer:_________________________
- Use the Integral Test to determine if the series 2
n n^^ ln n
∞
=
∑ converges or diverges.
Answer:_________________________
Problem 2
Find the radius of convergence and interval of convergence for the power series 1
n n n
x n
∞
=
Be sure to check any endpoints that exist.
Problem 3
(a) Find a power series representation for the function (^2)
f x x
. Then state the interval on which
the series equals the function.
(b) Use the series in (a) to find a power series representation for f ( ) x = arctan x = tan−^1 x.
Problem 5
a. Determine whether the sequence { a (^) n }=
n n
is increasing, decreasing, or not monotonic.
b. If 0
n n n
c x
∞
=
∑ is convergent when^ x^ = −^3 and divergent when^ x^ =^7 , is :
(a) 0
n^ ( 9) n n
c
∞
=
∑ − convergent?^ (b)^
0
n n
c
∞
=
∑ convergent?^ Give reasons for your answers.
