Calculus II 
Quiz #11

May 15, 2002

Name____________________ 
R. Hammack

Score ______

(1) Find the interval of convergence of the power series
Notice that this series will be alternating if x happens to be negative.
Therefore, we need to test for absolute convergence. Using the ratio test for
absolute convergence, we get
ρ =
Therefore ρ = and
the series will converge if ,
or if x< 4, or rather if 4
< x < 4.
What about the endpoints? If we plug x=4 into ,
we get the convergent pseries
If we plug x=4 into ,
we get the convergent alternating series.
Therefore, the interval of convergence is [4, 4].
(2) Find a power series representation of the function
We know: cos(u) =
Plugging in u=
to this, we get:
=
(3) Use your answer from part (2) to express as
an infinite series.
dx
=
=
Therefore dx =
(4) Consider the function f(x) =
. Find the formula for the Maclaurin
polynomial .
We know f(x) = ,
f '(x) =, f
''(x) =,
f '''(x) =,
... ,(x)
=,
...
and thus f(0) = 1,
f '(0) =, f
''(0) =1 , f '''(0) =, ... ,(0)
=,
...
Consequently, =