Sign of Bertrand

The sign of Bertrand ( de Morgan - Bertrand ) is a sign of the convergence of numerical series with positive members, established by Joseph Bertrand . A similar sign was obtained by Augustus de Morgan

Content

Wording

Row $\sum _{n=1}^{\infty }a_{n}$ ${\ displaystyle \ sum _ {n = 1} ^ {\ infty} a_ {n}}$ converges if $n>N$ ${\ displaystyle n> N}$ the inequality holds:
$B_{n}=\ln n\cdot \left(n\left({\frac {a_{n}}{a_{n+1}}}-1\right)-1\right)\geqslant \delta ,$ ${\ displaystyle B_ {n} = \ ln n \ cdot \ left (n \ left ({\ frac {a_ {n}} {a_ {n + 1}}} - 1 \ right) -1 \ right) \ geqslant \ delta,}$
Where $\delta >1$ ${\ displaystyle \ delta> 1}$ .
If $B_{n}\leqslant 1$ ${\ displaystyle B_ {n} \ leqslant 1}$ starting with some $n$ ${\ displaystyle n}$ then the series diverges.

Limit wording

If there is a limit:
$B=\lim _{n\to \infty }B_{n}$ ${\ displaystyle B = \ lim _ {n \ to \ infty} B_ {n}}$
then at $B>1$ ${\ displaystyle B> 1}$ the series converges, and when $B<1$ ${\ displaystyle B <1}$ - diverges.

Comment. If a $B=1$ ${\ displaystyle B = 1}$ , then the Bertrand trait does not answer the question of the convergence of the series.

The Bertrand sign is more sensitive than the Raabe sign and can be used for extremely slowly converging series.

Literature

Bertrand tag - article from the Mathematical Encyclopedia

Links

Weisstein, Eric W. Bertrand's Test on Wolfram MathWorld .
http://vuz.exponenta.ru/PDF/raabe.html