(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 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.

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,  =