Sign of Lobachevsky

The Lobachevsky sign is a sign of the convergence of the number series proposed by Lobachevsky between 1834 and 1836.

Let be $(a_{n})$ ${\ displaystyle (a_ {n})}$ $(a_ {n})$ there is a decreasing sequence of positive numbers, then the series
$\sum _{n=1}^{\infty }a_{n}$ ${\ displaystyle \ sum _ {n = 1} ^ {\ infty} a_ {n}}$ $\ sum _ {{n = 1}} ^ {\ infty} a_ {n}$
converges or diverges simultaneously with a number
$\sum _{m=1}^{\infty }{\frac {N_{m}}{2^{m}}}$ ${\ displaystyle \ sum _ {m = 1} ^ {\ infty} {\ frac {N_ {m}} {2 ^ {m}}}}$ $\ sum _ {{m = 1}} ^ {\ infty} {\ frac {N_ {m}} {2 ^ {m}}}$
Where $N_{m}$ ${\ displaystyle N_ {m}}$ $N_ {m}$ Is the smallest integer such that $a_{N_{m}}\leq {\frac {1}{2^{m}}}$ ${\ displaystyle a_ {N_ {m}} \ leq {\ frac {1} {2 ^ {m}}}}$ $a _ {{N_ {m}}} \ leq {\ frac 1 {2 ^ {m}}}$ .

Examples

For a harmonic series $a_{n}={\tfrac {1}{n}}$ ${\ displaystyle a_ {n} = {\ tfrac {1} {n}}}$ we have $N_{m}=2^{m}$ ${\ displaystyle N_ {m} = 2 ^ {m}}$ , in this way ${\frac {N_{m}}{2^{m}}}=1$ ${\ displaystyle {\ frac {N_ {m}} {2 ^ {m}}} = 1}$ and then the second row diverges. According to the sign of Lobachevsky, the first also diverges.

Literature

Lobachevsky sign - an article from the Mathematical Encyclopedia

Sign of Lobachevsky

Examples

Literature

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