206 Chapter 7Sequences and series
in whichs
ncontains 2
nterms of which the first, and largest, is 1 22
np. Each sum
s
nis then less than and
The series on the right is a convergent geometric series
∑
x
rwithx 1 = 1122
p− 11 < 11.
0 Exercises 46, 47
d’Alembert’s ratio test
8The seriesa
11 + 1 a
21 +1-1+ 1 a
r1 + 1 a
r+ 11 +1-
(i) converges if
(ii) diverges if
(iii) may do either if the limit equals 1, and further tests are necessary.
EXAMPLES 7.9The ratio test
(i) The geometric series 11 + 1 x 1 + 1 x
21 + 1 x
31 +1-
The general term isa
r1 = 1 x
rso thata
r+ 11 = 1 x
r+ 1and , independent of r. Then
and the series converges if|x| 1 < 11 , diverges if|x| 1 > 11 , and the test fails forx 1 = 1 ± 1.
In this case it is readily shown by inspection that the series diverges whenx 1 = 1 ± 1.
lim | |
r
a
a
x
rr→
+=
∞
1a
a
x
rr+=
1lim
r
a
a
rr→
+∞
11
lim
r
a
a
rr→
+<
∞
11
S
pp p<+
−− −1
1
2
1
2
1
2
121312
2
1
2
1nnpnp=
−8Jean LeRond d’Alembert (1717–1783). Secretary of the French Academy, he is best known for his contribution
to the post-Newtonian development of mechanics with his principle of virtual work(d’Alembert’s principle)
published in his Traité de Dynamique(1743). He contributed to the theory of partial differential equations, the
calculus, and infinite series. In his Différentiel(1754) in the Encyclopédie des sciences, des arts et des métiers
(1751–1772) he first gave the derivative as the limit of a quotient of increments but, because of the conceptual
difficulties associated with the limit as a process consisting of an infinite number of steps, the definition was not
accepted until the work of Cauchy (1821). He gave the complete solution of the precession of the equinoxes.
Although the ratio test is usually ascribed to him, and sometimes to Cauchy, it was probably first given in 1776 by
the Cambridge mathematician Edward Waring (1734–1793).