Example 1 Find the radiusof convergenceand intervalof convergenceof the series.
Solution.Let. Then as. So
the seriesis absolutelyconvergentfor |x| < 1 (and divergentfor |x| > 1) by the RatioTest.Rc= 1. This
leavesthe endpointvaluesto check.
If x = ± 1, then the seriesis absolutelyconvergentby thep-test.Hencethe seriesis absolutelyconvergent
for |x| ≤ 1. The intervalof convergenceis [-1,1].
Example 2 If the series converges,then is convergentatx= 2, i.e. 2 is insidethe in-
tervalof convergence.SoRc≥ 2. Conversely, if the series diverges,then is diver-
gent atx= -3, i.e. -3 is outsidethe intervalof convergence.SoRc≤ |- 3 | = 3.
Exercise
Find the radiusof convergenceand intervalof convergenceof the followingseries.
1.
2.
3.
4.
- Given convergesatx= 5 and divergesatx= -7. Deducewherepossible,the convergence
or divergenceof theseseries:
a.
b.
c.
d.