Mathematics of Physics and Engineering

Convergence 209

Proof. Define z = z — ZQ. For z ^ 0, apply the root test to the resulting
series (4.3.1):

K = limsup \anzn\^n = \z\ limsup |a„|1/n = \z\/R.

n n

If R = 0, then K — +oo for all z ^ 0, which means that the series diverges
for z 7^ 0. If R = +oo, then K = 0 for all 5, which means that the series
converges for all z. If 0 < R < +oo, then K < 1 for |z| < R and if > 1 for
\z\ > i?, so that the result again follows from the root test.
Representation (4.3.3) follows in the same way from the ratio test. •

EXERCISE 4.3.5.C Verify representation (4.3.3).

Definition 4.3 The number R introduced in Theorem 4.3.3 is called
the radius of convergence of the power series (4.3.1), and the set {z :
\z — zo| < R} is called the disk of convergence.

EXERCISE 4.3.6. c (a) Verify that the power series X)fc>o a*zk and
^fe>0 ka\izk have the same radius of convergence, (b) Give an example
of a power series that converges at one point on the boundary of the disk of
convergence, and diverges at another point. Explain why the convergence
in this example is necessarily conditional (in other words, explain why the
absolute convergence at one point on the boundary implies absolute conver-
gence at all points of the boundary).
In practice, it is more convenient to compute the radius of convergence
of a given series by directly applying the ratio test or the root test, rather
than by formulas (4.3.2) or (4.3.3). FOR EXAMPLE, to find the radius of
convergence of the power series

(-l)"z^3 "+1n!
n>l
f-ll"z3n+1n'
we write c„ = •—fan)" L anc

(2n)n (4.3.4)

where we used limn_>00(l + n-1)™ = e. The ratio test guarantees conver-
gence if |z|^3 /(2e) < 1 or \z\ < \/2e. Therefore, the radius of convergence is
\/2e. Direct application of either (4.3.2) or (4.3.3) is difficult because many
of the coefficients an in the series (4.3.4) are equal to zero.