Advanced High-School Mathematics

284 CHAPTER 5 Series and Differential Equations

5.3.1 Radius and interval of convergence

Our primary tool in determining the convergence properies of a power

series

∑∞

n=0

anxnwill be theRatio Test. Recall that the series

∑∞

n=0

|anxn|

will converge if

1 > nlim→∞

|an+1xn+1|

|anxn|

= |x|nlim→∞

∣∣

∣∣

∣

an+1

an

∣∣

∣∣

∣,

which means that

∑∞

n=0

anxnis absolutely convergent for all x satisfying |x|<nlim→∞

∣∣

∣∣

∣

an

an+1

∣∣

∣∣

∣.

The quantityR= limn→∞

∣∣

∣∣

∣

an

an+1

∣∣

∣∣

∣is sometimes called theradius of con-

vergenceof the power series

∑∞

n=0

anxn. Again, as long as−R < x < R,

we are guaranteed that

∑∞

n=0

anxn is absolutely convergent and hence

convergent.

A few simple examples should be instructive.

Example 1. The power series

∑∞

n=0

(−1)nxn

2 n+ 1

has radius of convergence

R = limn→∞

∣∣

∣∣

∣

an

an+1

∣∣

∣∣

∣ = limn→∞

( n

2 n+1

)

(n+1

2 n+3

) = limn→∞ n(2n+ 3)

(n+ 1)(2n+ 1)

= 1.

This means that the above power series has radius of convergence 1 and
so the series is absolutely convergent for− 1 < x <1.

Example 2. The power series

∑∞

n=0

nxn

2 n

has radius of convergence