11.10 Infinite Sequences & Series
Taylor & Maclaurin Series
Question #2 (Medium): Taylor Series and Its Radius of Convergence
( ( ) ( ) ( )
Taylor series states: ( ) ∑ ( ) ( ) ( ) ( )
Radius of convergence is determined using the ratio test | |
So any value of that makes the limit to be less than falls in the radius of convergence.
Find the Taylor series for ( ) centered at the given value of .
Assume that f has a power series expansion. Do not show that
Also find the associated radius of convergence.
( ) ,
Taylor series requires derivative in sequential order. Thus, for , use the function as it is, so
( ) ; for , its first derivative, matching with the number , so: ( ) ; for ,
second derivative: ( ) ; for , third derivative: ( )( ) ; for , fourth