(12.8) Power SeriesExample.For whatxdoes the series∑∞n=013nxn= 1 +13x+19x2+127x3+· · ·converge?This is just a geometric series. Recall∑∞n=0arnconverges toa1-rprovided|r|<1. Here,a= 1 andr=x/3. So the series converges to11-x/3=33-xprovided|x/3|<1, or|x|<3.That is, the series converges exactly when-3< x <3.We can think of this series as a function:f(x) = 1+13x+19x2+127x3+· · ·. This function hasdomain given by the interval (-3,3) because that’s where the series converges. Actually,f(x) =33-x, but with domain (-3,3).Example.For whatxdoes the series∑∞n=1xn2nn=12x+18x2+124x3+· · ·converge?We can use the ratio test. Here,an=xn2nn. When we do the ratio test, we take absolutevalues because depending on whatxwe choose, the series might not be positive. Solimn→∞an+1an= limn→∞xn+12n+1(n+ 1)xn2nn= limn→∞xn+12n+1(n+ 1)2nnxn= limn→∞nx2(n+ 1)=|x|2So we know the series converges if|x|2<1. (In fact, it converges absolutely in that case.)So it converges absolutely if|x|<2 or-2< x <2.We also know that it diverges if|x|2>1, or|x|>2. So it diverges ifx <-2 orx >2.