<abstract xmlns="http://www.w3.org/1999/xhtml">

It is well-known, as follows from the Stirling’s approximation <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0006_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>n</mi><mo>!</mo><mo>∼</mo><msqrt><mrow><mn>2</mn><mi>π</mi><mi>n</mi><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>n</mi><mo>/</mo><mi>e</mi></mrow><mo>)</mo></mrow></mrow></mrow><mi>n</mi></msup></mrow></msqrt></mrow></math>
<tex-math>n! \sim \sqrt {2\pi n{{\left( {n/e} \right)}^n}}</tex-math>
</alternatives>
</inline-formula>, that <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0006_eq_003.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mroot><mrow><mi>n</mi><mo>!</mo><mo>/</mo><mi>n</mi><mo>→</mo><mn>1</mn><mo>/</mo><mi>e</mi></mrow><mi>n</mi></mroot></mrow></math>
<tex-math>\root n \of {n!/n \to 1/e}</tex-math>
</alternatives>
</inline-formula>. A generalization of this limit is (11<italic>s</italic>· 22<italic>s</italic>· · · <italic>nns</italic>)1<italic>/ns</italic>+1 <italic>· n−</italic>1<italic>/</italic>(<italic>s</italic>+1) <italic>→ e−</italic>1<italic>/</italic>(<italic>s</italic>+1)2 which was established by N. Schaumberger in 1989 (see [8]). The aim of this work is to establish a new generalization that is in fact an improvement of Schaumberger’s formula for a general sequence <italic>An </italic>of positive real numbers. All of the results are applied to some well-known sequences in mathematics, for example, for the prime numbers sequence and the sequence of perfect powers.
</abstract>

It is well-known, as follows from the Stirling’s approximation 


n!∼2πn(n/e)n
n! \sim \sqrt {2\pi n{{\left( {n/e} \right)}^n}}

, that 


n!/n→1/en
\root n \of {n!/n \to 1/e}

. A generalization of this limit is (11s· 22s· · · nns)1/ns+1 · n−1/(s+1) → e−1/(s+1)2 which was established by N. Schaumberger in 1989 (see [8]). The aim of this work is to establish a new generalization that is in fact an improvement of Schaumberger’s formula for a general sequence An of positive real numbers. All of the results are applied to some well-known sequences in mathematics, for example, for the prime numbers sequence and the sequence of perfect powers.

A Further Generalization of <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2022-0006_eq_001.png"/>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mrow><mo>lim</mo></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mroot><mrow><mi>n</mi><mo>!</mo><mo>/</mo><mi>n</mi></mrow><mi>n</mi></mroot><mo>=</mo><mn>1</mn><mo>/</mo><mi>e</mi></mrow></math>
<tex-math>{\lim _{n \to \infty }}\root n \of {n!/n} = 1/e</tex-math>
</alternatives>
</inline-formula>

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution 4.0 International License.

It is well-known, as follows from the Stirling’s approximation 


n!∼2πn(n/e)n
n! \sim \sqrt {2\pi n{{\left( {n/e} \right)}^n}}

, that 


n!/n→1/en
\root n...

A Further Generalization of ${lim}_{n \to \infty} \sqrt[n]{n! / n} = 1 / e$ {\lim _{n \to \infty }}\root n \of {n!/n} = 1/e

Publié en ligne: 18 avr. 2022

Pages: 167 - 175

Reçu: 08 juil. 2021

Accepté: 22 mars 2022

DOI: https://doi.org/10.2478/amsil-2022-0006

Mots clés
limit formula, generalization, prime numbers, perfect powers

© 2022 Reza Farhadian et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

A Further Generalization of limn→∞n!/nn=1/e {\lim _{n \to \infty }}\root n \of {n!/n} = 1/e

Publié en ligne: 18 avr. 2022

Pages: 167 - 175

Reçu: 08 juil. 2021

Accepté: 22 mars 2022

DOI: https://doi.org/10.2478/amsil-2022-0006

Mots cléslimit formula, generalization, prime numbers, perfect powers

© 2022 Reza Farhadian et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

A Further Generalization of ${lim}_{n \to \infty} \sqrt[n]{n! / n} = 1 / e$ {\lim _{n \to \infty }}\root n \of {n!/n} = 1/e

Mots clés
limit formula, generalization, prime numbers, perfect powers