CK12 Calculus - Single Variable

6. For whatvaluesofcis the series convergent?

PowerSeries

PowerSeriesand Convergence
Definition(PowerSeries)
APowerSeriesis a seriesof the form

(PS1)

wherexis a variableand thean's are constants(in our case,real numbers)calledthecoefficientsof the
series.
The summationsign ∑ is a compactand convenientshorthandnotation.Readerunfamiliarwith the notation
mightwantto writeout the detaila few timesto get usedto it.
Powerseriesare a generalizationof polynomials,potentiallywith infinitelymanyterms.As observed,the

indicesnofanare non-negative,so no negativeintegralexponentsofx, e.g. appearsin a powerseries.

Moregenerally, a seriesof the form

(PS2)

is calleda powerseriesin (x-x 0 ) (1 = (x-x 0 )^0 ) or a powerseriescenteredatx 0 ((PS1)representsseries
centeredatx= 0).
Givenany valueofx, a powerseries((PS1)and (PS2))is a seriesof numbers.The first questionis:
Is the powerseries(as in (PS1)or (PS2))a functionofx?
Sincethe seriesis alwaysdefinedatx= 0 (resp.x=x 0 ), the questionbecomes:

For whatvalueofxis a powerseriesconvergent?
The answersare knownfor someseries.Convergencetestscouldbe appliedon someothers.
Example 1 Let r ≠ 0 andx 0 be real.

is absolutelyconvergentand

equals for , i.e. , and divergesotherwise.