CK-12-Calculus

8.6. Power Series http://www.ck12.org

2. Discuss the convergence of the series∑∞n= 02 nx^2 n.
   Hint: Apply a combination of tests in §8.5.

Interval and Radius of Convergence

The following theorem characterizes the values ofxwhere a power series is convergent.
Theorem(Interval of convergence)
Given a power series∑∞n= 0 an(x−x 0 )n. Exactly one of the following three describes all the values where the series
is convergent:
(A) The series converges exactly atx=x 0 only.
(B) The series converges for allx.
(C) There is a real numberRc>0 that the series converges if|x−x 0 |<Rcand diverges if|x−x 0 |>Rc.
ThisRcis unique for a power series, called theradius of convergence. By conventionRc=0 for case (A) andRc=∞
for case (B). The only two values ofxthe Theorem cannot confirm are theendpoints x=x 0 ±Rc. In any case, the
valuesxwhere the series converges is an interval, called theinterval of convergence. It is the singletonx 0 for case
(A) and(−∞,∞)for case (B). For case (C), it is one of the four possible intervals:(x 0 −Rc,x 0 +Rc),(x 0 −Rc,x 0 +
Rc],[x 0 −Rc,x 0 +Rc), and[x 0 −Rc,x 0 +Rc]. Here, the endpoints must be checked separately for convergence.
Example 1Find the radius of convergence and interval of convergence of the series∑∞n= 0 xnn 2.

Solution. Letbn=xnn 2. Then

∣∣

∣bnb+n^1

∣∣

∣=

∣∣

∣(nx+n+ 11 ) 2 ·nxn^2

∣∣

∣=

∣∣

∣∣( 1 +^11

n

) 2

x

∣∣

∣∣→|x|asn→∞. So the series is absolutely convergent

for|x|<1 (and divergent for|x|>1) by the Ratio Test.Rc=1. This leaves the endpoint values to check.
Ifx=±1, then the series is absolutely convergent by thep−test. Hence the series is absolutely convergent for
|x|≤1. The interval of convergence is[− 1 , 1 ].
Example 2If the series∑∞n= 0 an 2 nconverges, then∑∞n= 0 anxnis convergent atx=2, i.e. 2 is inside the interval of
convergence. SoRc≥2. Conversely, if the series∑∞n= 0 an(− 3 )ndiverges, then∑∞n= 0 anxnis divergent atx=−3, i.e.
−3 is outside the interval of convergence. SoRc≤|− 3 |= 3.
Exercises
Find the radius of convergence and interval of convergence of the following series.

1.∑∞n= 1 nxn

2.∑∞n= 1 (−^1 n)nxn

3.∑∞n= 1 xnn/!^3

4.∑∞n= 1 √n(x−x 0 )n

5. Given∑∞n= 0 anxnconverges atx=5 and diverges atx=−7. Deduce where possible, the convergence or
   divergence of these series:
   a.∑∞n= 0 an
   b.∑∞n= 0 an 3 n
   c.∑∞n= 0 an(− 8 )n
   d.∑∞n= 0 an( 9 )n
   e.∑∞n= 0 an( 6 )n