ABSTRACT
The proof of the absolute convergence [Equations 10.387a through 10.387d (Equations 10.392a through 10.392d)] of the series [Equations 10.375a and 10.375c (Equations 10.391a and 10.391c)] used the D’Alembert ratio test (Subsection I.29.3.2) that specifies a sufficient condition, namely, if the ratio of successive terms in modulus tends to less than unity, the series is absolutely convergent (Equation I.29.31b). This sufficient condition is not necessary, as can be shown by counter example(s) of convergent series whose ratio of successive terms does not tend to less than unity and may even diverge. To construct such counter example to the ratio test, it is sufficient to mix two series with distinct rates of convergence, for example, alternating the terms of two geometric (Equation 10.393b) [harmonic (Equation 10.394b)] series:
> > = + + + + = + =
∞∑b a f a b a b a bn n m
: ( ), (10.393a and 10.393b)
Re( ) Re( ) :β α β α β α β α> > = + + + + = +
1
1 1
1 1
1 2
1 2
1 1 g
n n
(10.394a and 10.394b)
with distinct arguments (Equation 10.393a) [exponents (Equation 10.394a)]. Both series (Equations 10.393b and 10.394b) are absolutely convergent because the series of moduli can be summed exactly (Equations 10.395 and 10.396, respectively):
a b a
a
b
b
+( ) = −
+ −
1 1 , (10.395)
n n n n n n
+( ) = + =∑ ∑β α β α ζ 1 1
Re Re (Re(β ζ α)) (Re( )),+ (10.396)
where the sum of the geometric series (Equations I.21.62a and I.21.62b ≡ Equations 3.148a and 3.148b) [the zeta-function (Equations 10.386a and 10.386b)] was used. Besides
n n n
n
α
≡ = =
exp log( ) exp( log )
exp [Re( )log ] exp[i n
n n
Im( )log ]
exp[Re( )log ] Re
α
α α
= = ( )
(10.397)
in Equation 10.396, Equation 10.397 was used in agreement with Equation 3.76b.
