Series of Bertrand

For $\ alpha$ and $\ beta$ two realities, one calls series of Bertrand the series with real terms positive following:

$\ sum_ {N \ Ge 2} {1 \ over n^ \ alpha \, (\ ln N) ^ \ beta}$ .

The harmonic Série is a particular case (with the first term near), for $\ alpha = 1$ and $\ beta = 0$ :

\ sum_ {N \ Ge 2} {1 \ over N} = \ frac {1} {2} + \ frac {1} {3} + \ frac {1} {4} +… + \ frac {1} {N} +…

This requirement and sufficient is sometimes summarized in: “the couple $(\ alpha, \ beta)$ is lexicographiquement posterior with $(1,1)$ ”. That refers to the order adopted to sort the words in a dictionary: one takes account of the first letter, then second, etc

Example:

It is known that the harmonic Série diverges, and that the series $\ sum_ {N \ Ge 1} {1 \ over n^ \ alpha}$ converges if and only if $\ alpha > 1$ (Série of Riemann). Well-sure, if $\ alpha > 1$ , ${1 \ over n^ \ alpha} =o \ left (\ frac {1} {N} \ right)$ , however there exist series whose general term is negligible in front of $\ frac {1} {N}$ but which diverges. The series of Bertrand then allows exhiber a counterexample the following erroneous implication: $(u_n) _ {N \ in \ NR} =o \ left (\ frac {1} {N} \ right) \ Rightarrow \ sum_ {N \ in \ NR} u_n$ converges. “(1,1) - series of Bertrand”, IE the series:

\ sum_ {N \ Ge 2} {1 \ over N \, \ ln N}

is divergent, according to the proposal, whereas

{1 \ over N \, \ ln N} =o \ left (\ frac {1} {N} \ right)

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