Logarithmic convergence criterion

The logarithmic convergence criterion is a convergence criterion in analysis , one of the branches of mathematics . There are sufficient conditions for both the convergence and the divergence of series whose terms form a sequence of positive real numbers .

Formulation of the criterion

The criterion says the following:

Be a sequence of numbers in and be any number  . ${\displaystyle (a_{k})_{k\in \mathbb {N} }}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle a_{k}>0}$   ${\displaystyle (k\in \mathbb {N} )}$

It is assumed that the sequence of numbers formed for this purpose   with ${\displaystyle (b_{k})_{k\in \mathbb {N} }}$

${\displaystyle b_{k}={\frac {\ln(k\cdot a_{k})}{\ln(\ln(k))}}\;(k\in \mathbb {N} )}$

actually or improperly converge and thereby the limit value

${\displaystyle L=\lim _{k\to \infty }{b_{k}}\in [-\infty ,\infty ]}$

have.

Then:

(I) In the case     the corresponding series is convergent : ${\displaystyle L<-1}$ ${\displaystyle {\sum }_{k}{a_{k}}}$

${\displaystyle \sum _{k=1}^{\infty }{a_{k}}<\infty }$   .

(II) In the case     , the corresponding row is divergent : ${\displaystyle L>-1}$ ${\displaystyle {\sum }_{k}{a_{k}}}$

${\displaystyle \sum _{k=1}^{\infty }{a_{k}}=\infty }$   .

Notes on evidence

The proof is based on the major and minor criterion and on that the series

${\displaystyle \sum _{k=2}^{\infty }{\frac {1}{k\cdot {\ln(k)}^{1+\delta }}}}$

for     converges and for     diverges. ${\displaystyle \delta >0}$ ${\displaystyle \delta =0}$

For the case of convergence, the integral criterion and the fact that then

${\displaystyle \textstyle \int _{2}^{\infty }{\frac {1}{t\cdot {\ln(t)}^{1+\delta }}}\,\mathrm {d} t={\frac {1}{\delta \cdot {\ln(2)}^{\delta }}}}$

is.

application

• For

${\displaystyle a_{k}={\frac {1}{k^{2}}}\;\;(k\in \mathbb {N} )}$

one has

${\displaystyle \lim _{k\to \infty }{b_{k}}=\lim _{k\to \infty }{\frac {\ln({\frac {1}{k}})}{\ln(\ln(k))}}=\lim _{k\to \infty }{\frac {-\ln(k)}{\ln(\ln(k))}}=\lim _{x\to \infty }{\frac {-x}{\ln(x))}}=-{\infty }<1}$  ,

which, according to the criterion, is evidence of the convergence of the known series

${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}=1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+\cdots }$

represents.

• For

${\displaystyle a_{k}={\frac {1}{k}}\;\;(k\in \mathbb {N} )}$

one has

${\displaystyle \lim _{k\to \infty }{b_{k}}=\lim _{k\to \infty }{\frac {\ln(1)}{\ln(\ln(k))}}=0>-1}$  ,

with which the criterion proves the divergence of the harmonic series .

annotation

  No statements regarding convergence or divergence can be made about the "case of doubt"   . I.e. depending on the sequence of numbers presented, both cases can occur. ${\displaystyle \lim _{k\to \infty }{b_{k}}=-1}$

literature

• Kazimierz Kuratowski : Introduction to Calculus (=  International Series of Monographs in Pure and Applied Mathematics . Volume 17 ). 2nd Edition. Pergamon Press, Oxford et al. a. 1969 ( MR0349918 ).

References and comments

1. ↑ Kazimierz Kuratowski : Introduction to Calculus (=  International Series of Monographs in Pure and Applied Mathematics . Volume 17 ). 2nd Edition. Pergamon Press, Oxford et al. a. 1969, p. 298-299, 329 ( MR0349918 ).
2. ↑ Kazimierz Kuratowski : Introduction to Calculus (=  International Series of Monographs in Pure and Applied Mathematics . Volume 17 ). 2nd Edition. Pergamon Press, Oxford et al. a. 1969, p. 296-297, 298-299 ( MR0349918 ).