RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

11.4. Means of testing for convergence of series 213
Thus, the tail of the series was term-wise less than a convergent geometric progression,
and the convergence of the series ∑ un was concluded. Similarly, CAUCHY proved the
divergence part of the root test by comparing with a divergent geometric progression.
CAUCHY based his proof of the ratio test on the following result which he had
previously obtained.
“2nd theorem. If the function f (x) is positive for large values of x and the
ratio
f (x + 1)
f (x)
converges toward the limit k when x increases indefinitely, the expression
[ f (x)] 1 x
will converge at the same time toward the same limit.” 15
In his proof, CAUCHY distinguished between two cases namely k finite or not; we
shall here only be concerned with the case where k is finite. CAUCHY let ε denote an
as yet unspecified number which was presumably very small. By assuming that for
x ≥ h,
f (x + 1)
f (x)
∈ [k − ε, k + ε] ,
CAUCHY found by the theory of means which he developed in a note 16 that the geo-
metric mean 17 of
f (h + 1)
,
f (h)
f (h + 2)
, . . . ,
f (h + 1)
f (h + n)
f (h + n − 1) ,
would also belong to this interval, i.e.
�
n f (h + n)
= k + α, α ∈ [−ε, ε] .
f (h)
Then, CAUCHY found by inserting x = h + n
which meant
f (x) = f (h) · (k + α) x−h
f (x) 1 x = f (h) 1 x · (k + α) 1− h x →
x→∞ k + α.
15 “2.e Théorème. Si, la fonction f (x) étant positive pour de très-grandes valeurs de x, le rapport
f (x + 1)
f (x)
converge, tandis que x croit indéfiniment, vers la limite k, l’expression
[ f (x)] 1 x
convergera en même temps vers la même limite.” (ibid., 53–54).
16 (ibid., note II).
17 The geometric mean of the quantities a1, . . . , an was the quantity n� ∏ n k=1 a k.