11.8 Infinite Sequences & Series
Question #1 (Easy): Determining the Radius and Interval of Convergence
Power series contain variable in the series expression so that ∑
where ’s are coefficients and is a variable. Power series looks like polynomial, except with
infinite terms. Sometimes restrictions apply to the possible values that can take to make the power
series convergent. Ratio test can be used to determine this radius for the series convergence. Then
various series tests learned so far can apply to be more specific about the interval of convergence.
Find the radius of convergence and interval of convergence of the series.
Notice that the series contains a variable x. Thus, the range of values that x can take which makes the
series convergent need to be determined. In order to do so, first put into the ratio test and get the
outcome of the ratio test which will contain the variable x. Then the value of x can be determined which
will make outcome of the ratio test to be less than 1.
Then and ( ) . The ratio test: | | | | |
( ) ( )