If and convergeson a commoninterval|x| <R ab, thentheir
productpowerseries also convergesonR aband is the powerseriesfor the productfunction
f(x)g(x).
Example 1 Find a powerseriesfor.
Solution. with radiusof convergence1 and with radiusof convergence
.So for ,
Exercise
- Find the first 4 termsof a powerseriesfor.
- Find a powerseriesfor wherer,s> 0 are real numbersand
(i)r≠s
(ii)r=s.
Taylorand MaclaurinSeries
Taylorand MaclaurinPolynomials
We knowthe linearapproximationfunctionL 1 (x) to a (smooth)functionf(x) atx=x 0 is givenby the tangent
line at the point.L 1 (x 0 ) = f(x 0 ) and. Indeed,this is the only linearfunctionwith these 2
properties.