Exercise
- Find a powerseriesrepresentationof atx= 0.
- Find a powerseriesrepresentationof atx= 0.
- Notice. Find a powerseriesrepresentationof at
. In whatintervalis the equalitytrue?
ChoosingCenters
TaylorSeries(indeedTaylorpolynomialsof lowerdegrees)oftenprovidegoodapproximationof functions.
However, the choiceof centercoulddetermine
(1) whetherthe intendedvalueofxis insidethe intervalof convergence
(2) rate of convergence,i.e. how manytermsto take to achieveprescribeddegreeof accuracy
For frequentlyusedfunctions,the first choicemay be the standardcenter(see the list at the end of this
section).
Example 1 Approximateln0:99
Solution.Since.99 is closeto the centerx= 1, we use the standardTaylorseriesforln(1 -x).
Thenwe may be able to deducea usefulTaylorSeriescenteredcloseto the givenx.
Example 2 Approximatesin(1.1) to 4 decimalplaces.
Since1.1 is closeto , we wouldtry to find a TaylorSeriesofsin xat. Letf(x) =sin x. Then