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Cauchy-Bolzano convergence criterion for infinite series

Theorem (Cauchy-Bolzano convergence criterion): The infinite series

(1)

holds for all and all .

In other words, the series is convergent if and only if the sequence of its partial sums

is Cauchy .

As it is often the case Cauchy was not the first mathematician who derived it. A similar form of the criterion can be found in the work of  Lacroix, Euler and da Cunha several years before Cauchy. Cauchy writes in his Cours d'Analyse (1821): ... il est nécessaire et il suffit que la différence

devienne infiniment petite, quand on attribue au nombre une valeur infiniment grande, quel que soit d'ailleurs le nombre entier représenté par '' [  [1]   or  [2] , vol.7, p. 267].  It can also be found in Bernard Bolzano’s ¹   Rein analytischer Beweis (1817), where he presented a foundation of the differential calculus free from the concept of infinitesimals  [3]  .  

Example: Consider the alternating harmonic series

(2)

To apply the above criterion it is sufficient to prove that

(3)

for every .

The left hand inequality can be verified easily. It is enough to group the summands in to groups of two successive terms. Every such group is actually a positive number. If is even the result is a sum of positive numbers, if is odd the alone standing term is positive, and the conclusion follows. To prove the right hand inequality group the terms in the following way  

Since the value of every difference in the parentheses is positive we always decrease the first term by a non-negative value independently whether is odd or even, and (3) is proved.

Corollary: If (1) is a convergent series then .

Notes

References

[1]	Cauchy, A. L. (1821). Cours d'Analyse de l'École Royale Polytechnique: Première Partie: Analyse Algébrique . Paris: Chez Debure frères.

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