1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

4.3 • GEOMETRIC SERIES AND CONVERGENCE T HEOREMS 143

oo n

• EXAMPLE 4.14 Evaluate L (~).

n =3

Solution We can put this expression in the form of a geometric series:

=

=

(by Equation (4- 10) in T heorem 4.5)

(by reindexiog)

(by Theorem 4. 12 because l~I = 4 < 1)

(by standard simplillcation procedures).

Remar k 4.3 The equality given in Example 4.14 illustrates an important point

with regard to evaluating a geometric series whose beginning index is other than

00 r

zero. The value of L zn equals 1 :,. If we think of z as the "ratio" by which

n=r

any term of the series is multiplied to generate successive terms, we note that

the sum of a geomet ric series equals fif~r~~~:, provided I ratio I < 1. •

The geometric series is used in the proof of Theorem 4.13, which is known

as the ratio test. It is one of the most commonly used tests for determining

the convergence or divergence of series. The proof is similar to the one used for

real series, and we leave it for you to do.

00 /j_O\n

• EXAMPLE 4.15 Show that L ~ converges.
  n=O
  Solut ion Using the ratio test, we find that
  . l(1-it+

1

/(n+l)!I. n!ll-il. 11 - il

Jim n = hm = lrm--

n -oo 1 (1 -i) /n!I n - oo (n + 1)! n-oo n + 1

= lim y'2 = 0 = L.

n-oo n + 1

Because L < 1, the series converges.