Gyula Bereznai

Quote from the publication A Simple Convergence Criterion:[2]

Theorem: If there is a real ${\displaystyle \sum _{}a_{n}}$ for a positive numeric string ${\displaystyle p>e}$ and a natural ${\displaystyle N}$ such that every time ${\displaystyle n>N}$, each time

${\displaystyle \left({\frac {a_{n}}{a_{n+1}}}\right)^{n}\geq p}$,

then the series is convergent. And if

${\displaystyle \left({\frac {a_{n}}{a_{n+1}}}\right)^{n}\leq e}$,

then the series ${\displaystyle \sum _{}a_{n}}$ is divergent.

Thus Bereznai's theorem is:

Let ${\displaystyle \left(a_{n}\right)}$ be a sequence of positive numbers, such that there exists ${\displaystyle p\in \mathbb {R} }$

with ${\displaystyle \left({\frac {a_{n}}{a_{n+1}}}\right)^{n}\geq p>e\ (n=1,2,...)}$. Then the series ${\displaystyle \sum _{n=1}^{\infty }{a_{n}}}$ is convergent.

If ${\displaystyle \left({\frac {a_{n}}{a_{n+1}}}\right)^{n}\leq e}$, then the series is divergent.

It is known that Gyula Bereznai's method is more effective than the so-called D'Alembert quotient, which is most commonly used to decide the positive convergence of numerical lines and the so-called Raabe-Duhamel Method. That is, Gyula Bereznai's result, among other things, provides a useful tool for the study of a very intensively researched branch of mathematics, harmonic analysis. (Dr. Habil. György Gát)[3]