Cauchy's Criterion for Convergence

Augustin Cauchy, 

Many mathematicians of the day believe that your largest accomplishments were in the field of complex functions. You are known for your book that you mentioned in your second letter called Cours d'Analyse, where you stress the importance of "rigor" in analysis. It is in this book where you defined Cauchy's Criterion for Convergence. Since your book is written in French, I will recite a translated version. It states, "[it] is that a necessary and sufficient condition for convergence of a sequence {xn} is that it be a Cauchy sequence, where a Cauchy sequence is defined to be for each e > 0, there is a positive integer N such that | xn - xm | < e for all m > N and all n > N." (Cauchy). It is extremely difficult to find what value N the series converges to, and you realized that it is not necessary to find the best or most exact value for N. You need only find any value that works.

You were not the first mathematician to come up with similar notions, but because of your numerous accomplishments in this field, the name Cauchy has attached itself. Another reason your name may still be connected to this result is because of the definitions you included for convergence and divergence. The book states, "Let sn = u0 + u1 + u2 + ... + un-1 be the sum of the first n terms [of a series]..." In this case, n is any integer. The book continues by stating that for increasing values of n, if sn, which is the sum, indefinitely approaches a limit, then the series is convergent, and it converges to the limit, which is sn. If, as n increases, the sum does not indefinitely approach a certain limit, then the series is divergent, so there is no sn. This means that there is no sum.  

Then, you stated your Criterion for Convergence. You said that convergence is only possible if the sums un + un-1, un + un+1 + un+2, un + un+1 + un+2 + un+3,... " finish by constantly having an absolute value less than any assignable limit"(Cauchy). You did not prove this, but it is because the math needed to prove it was not "invented" yet. In the 1800's there was no definition of a real number. You did, however, provide examples of when your criterion holds for convergence. One of your examples is as follows: the expression 1 + x + x^2 + ... for | x | < 1 converges and satisfies your criterion because by substituting these values into un + un-1, un + un+1 + un+2, un + un+1 + un+2 + un+3,..., we get x^n, x^n + x^(n+1),... This expression is always between the bounds x^n and x^n/(1-x). You showed this in Cours d'Analyse and actually computed the values. As you can see, you have made extremely important contributions to the study of mathematics, and your legacy will continue to live on for many years to come. 

Regan Kapalko

Chaitesipaseut, S. (n.d.). Retrieved March 13, 2018, from https://math.berkeley.edu/~robin/Cauchy/index.html

Cauchy criteria. (n.d.). Retrieved March 14, 2018, from https://www.encyclopediaofmath.org/index.php/Cauchy_criteria