Math 110 Exam 3
November 16, 2000
Name
1. Determine whether the following statements are true or false. If true, explain why.
If false, explain why or give an example that disproves the statement. Your
explanations are worth more than your answers.
A) If lim a n  0 , then
n 
a
n
converges.
B) The Ratio Test can be used to determine whether
C) If 0  a n  bn and
b
n
diverges, then
a
n
1
n
3
converges.
diverges.
D) If {a n } and {bn } are divergent, then {a n  bn } is divergent
2. Use a test to determine whether the following series converge or diverge. If a
series is convergent, find the exact value of the sum, or find S 20 , the 20th partial
sum.

n
A)  3
n 1 n  1
B) 4 
8 16 32


 ...
5 25 125
3. A) Find the radius of convergence and the interval of convergence for the power

( x  7) n
series 
n
n 1
B) For what values of x does the power series 1  2 x  4 x 2  8 x 3  ... converge? What
does it converge to? Illustrate your result with graphs.
4. A) Derive the Maclaurin series for f ( x)  cos( x) . Write out at least four nonzero
terms, and also write it in summation notation.
B) Assume your series above converges for all x. Replace x by x 2 to obtain the
Maclaurin series for cos( x 2 )
C) cos( x 2 ) does not have an elementary antiderivative. However, we can approximate
b
 cos( x
2
)dx using our Maclaurin series. Use the first three nonzero terms to approximate
a
0.5
 cos( x
2
)dx . How big an error might we have made, according to the error analysis for
0
alternating series?