May 15, 2002
(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
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 p-series
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.
Therefore dx =
(4) Consider the function f(x) =
. Find the formula for the Maclaurin
We know f(x) = , f '(x) =, f ''(x) =, f '''(x) =, ... ,(x) =, ...
and thus f(0) = 1, f '(0) =, f ''(0) =1 , f '''(0) =, ... ,(0) =, ...