Mathematics of Physics and Engineering

Series of Complex Numbers 207

The following result is known as the ratio test for convergence.

Theorem 4.3.1 Define

„ ,. Icn+l| T v • c lc«+ll

K = hm sup ^-j—j-, L — hm inf ——p.

n jCn| n |Cn|

Then the series J2k>o Cfc

• Converges absolutely if K < 1;


• Diverges if L > 1;


• Can either converge or diverge, if K > 1 or L < 1.

Proof. If K < 1, then, by the definition of limsup, there exists a q G (K, 1)
and a positive integer N so that |cn+1| < q\cn\ for all n > AT. Then
|c;v+fc| < qk\cN\, k>l, and
AT-l JV-l , ,

fc>o fe=o fc>i fc=o q

If L > 1, then, by the definition of liminf, there exists a q € (1,L) and
a positive integer N so that |cn+i| > q\cn\ for all n > N. Therefore,
limn^oo \cn\ ^ 0, and the series diverges.
The following exercise completes the proof of the theorem. •

EXERCISE 4.3.3. B Construct three sequences J2k>ock > 3 — 1>2,3, so

t/iai 2Zfc>o cfe converges absolutely, X]fc>o cfc converges conditionally, and

Sfc>o cfc diverges, while the ratio test in all three cases gives K = 2 and

L = 0.

The following result is known as the root test.

Theorem 4.3.2 Define

M = limsup |cn|1/n.

n

Then the series J2k>0 ck

• Converges absolutely if M < 1;
  
  
  • Diverges if M > 1;
  
  
  
  


• Can either converge or diverge, if M = 1.