$(frac{n}{e})^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}$ doesn't comply with the limit definition?












2












$begingroup$


I try to understand what I've overlooked, when I came up with this inequality:



First, we have this limit:
$$limlimits_{n to infty} sqrt[n]{frac{n!}{n^n}} = frac{1}{e}$$
Which gives, by the definition of limit and some simple transformations:



$frac{1}{e} - varepsilon < sqrt[n]{frac{n!}{n^n}} < frac{1}{e} + varepsilon$



$(frac{n}{e} - nvarepsilon)^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}quadforall varepsilon > 0$



Then, we have this well-known inequality (multiple proofs can be found on math.stackexchange):
$$(frac{n}{e})^{n} < n!$$
So we have:
$$(frac{n}{e})^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}$$
According to this inequality, we cannot make $varepsilon$ arbitrary small, which contradicts the definition of limit.
What am I missing here?






sequences-and-series limits factorial







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What do you mean by $exp$? As far as I know, it's meant to be a function, but you're not giving it an input.
    $endgroup$
    – Calvin Godfrey
    Jan 10 at 20:39










  • $begingroup$
    @CalvinGodfrey, they just mean $e$.
    $endgroup$
    – Joe
    Jan 10 at 20:40






  • 3




    $begingroup$
    @Joe In that case $exp(1)$ would be more accurate.
    $endgroup$
    – cansomeonehelpmeout
    Jan 10 at 20:41










  • $begingroup$
    @cansomeonehelpmeout, yes that would have been more appropriate. I'm just explaining what OP meant. I absolutely do not condone their notation.
    $endgroup$
    – Joe
    Jan 10 at 20:43










  • $begingroup$
    @mfl, thanks, I was stuck because I didn't consider that all 3 of expressions in this inequality are sequences and thought in terms of fixed n
    $endgroup$
    – dpd
    Jan 10 at 21:02
















2












$begingroup$


I try to understand what I've overlooked, when I came up with this inequality:



First, we have this limit:
$$limlimits_{n to infty} sqrt[n]{frac{n!}{n^n}} = frac{1}{e}$$
Which gives, by the definition of limit and some simple transformations:



$frac{1}{e} - varepsilon < sqrt[n]{frac{n!}{n^n}} < frac{1}{e} + varepsilon$



$(frac{n}{e} - nvarepsilon)^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}quadforall varepsilon > 0$



Then, we have this well-known inequality (multiple proofs can be found on math.stackexchange):
$$(frac{n}{e})^{n} < n!$$
So we have:
$$(frac{n}{e})^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}$$
According to this inequality, we cannot make $varepsilon$ arbitrary small, which contradicts the definition of limit.
What am I missing here?






sequences-and-series limits factorial







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What do you mean by $exp$? As far as I know, it's meant to be a function, but you're not giving it an input.
    $endgroup$
    – Calvin Godfrey
    Jan 10 at 20:39










  • $begingroup$
    @CalvinGodfrey, they just mean $e$.
    $endgroup$
    – Joe
    Jan 10 at 20:40






  • 3




    $begingroup$
    @Joe In that case $exp(1)$ would be more accurate.
    $endgroup$
    – cansomeonehelpmeout
    Jan 10 at 20:41










  • $begingroup$
    @cansomeonehelpmeout, yes that would have been more appropriate. I'm just explaining what OP meant. I absolutely do not condone their notation.
    $endgroup$
    – Joe
    Jan 10 at 20:43










  • $begingroup$
    @mfl, thanks, I was stuck because I didn't consider that all 3 of expressions in this inequality are sequences and thought in terms of fixed n
    $endgroup$
    – dpd
    Jan 10 at 21:02














2












2








2





$begingroup$


I try to understand what I've overlooked, when I came up with this inequality:



First, we have this limit:
$$limlimits_{n to infty} sqrt[n]{frac{n!}{n^n}} = frac{1}{e}$$
Which gives, by the definition of limit and some simple transformations:



$frac{1}{e} - varepsilon < sqrt[n]{frac{n!}{n^n}} < frac{1}{e} + varepsilon$



$(frac{n}{e} - nvarepsilon)^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}quadforall varepsilon > 0$



Then, we have this well-known inequality (multiple proofs can be found on math.stackexchange):
$$(frac{n}{e})^{n} < n!$$
So we have:
$$(frac{n}{e})^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}$$
According to this inequality, we cannot make $varepsilon$ arbitrary small, which contradicts the definition of limit.
What am I missing here?






sequences-and-series limits factorial







share|cite|improve this question











$endgroup$




I try to understand what I've overlooked, when I came up with this inequality:



First, we have this limit:
$$limlimits_{n to infty} sqrt[n]{frac{n!}{n^n}} = frac{1}{e}$$
Which gives, by the definition of limit and some simple transformations:



$frac{1}{e} - varepsilon < sqrt[n]{frac{n!}{n^n}} < frac{1}{e} + varepsilon$



$(frac{n}{e} - nvarepsilon)^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}quadforall varepsilon > 0$



Then, we have this well-known inequality (multiple proofs can be found on math.stackexchange):
$$(frac{n}{e})^{n} < n!$$
So we have:
$$(frac{n}{e})^{n} < n! < (frac{n}{e} + nvarepsilon)^{n}$$
According to this inequality, we cannot make $varepsilon$ arbitrary small, which contradicts the definition of limit.
What am I missing here?






sequences-and-series limits factorial




sequences-and-series limits factorial






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 10 at 20:49







dpd

















asked Jan 10 at 20:36









dpddpd

134




134








  • 1




    $begingroup$
    What do you mean by $exp$? As far as I know, it's meant to be a function, but you're not giving it an input.
    $endgroup$
    – Calvin Godfrey
    Jan 10 at 20:39










  • $begingroup$
    @CalvinGodfrey, they just mean $e$.
    $endgroup$
    – Joe
    Jan 10 at 20:40






  • 3




    $begingroup$
    @Joe In that case $exp(1)$ would be more accurate.
    $endgroup$
    – cansomeonehelpmeout
    Jan 10 at 20:41










  • $begingroup$
    @cansomeonehelpmeout, yes that would have been more appropriate. I'm just explaining what OP meant. I absolutely do not condone their notation.
    $endgroup$
    – Joe
    Jan 10 at 20:43










  • $begingroup$
    @mfl, thanks, I was stuck because I didn't consider that all 3 of expressions in this inequality are sequences and thought in terms of fixed n
    $endgroup$
    – dpd
    Jan 10 at 21:02














  • 1




    $begingroup$
    What do you mean by $exp$? As far as I know, it's meant to be a function, but you're not giving it an input.
    $endgroup$
    – Calvin Godfrey
    Jan 10 at 20:39










  • $begingroup$
    @CalvinGodfrey, they just mean $e$.
    $endgroup$
    – Joe
    Jan 10 at 20:40






  • 3




    $begingroup$
    @Joe In that case $exp(1)$ would be more accurate.
    $endgroup$
    – cansomeonehelpmeout
    Jan 10 at 20:41










  • $begingroup$
    @cansomeonehelpmeout, yes that would have been more appropriate. I'm just explaining what OP meant. I absolutely do not condone their notation.
    $endgroup$
    – Joe
    Jan 10 at 20:43










  • $begingroup$
    @mfl, thanks, I was stuck because I didn't consider that all 3 of expressions in this inequality are sequences and thought in terms of fixed n
    $endgroup$
    – dpd
    Jan 10 at 21:02








1




1




$begingroup$
What do you mean by $exp$? As far as I know, it's meant to be a function, but you're not giving it an input.
$endgroup$
– Calvin Godfrey
Jan 10 at 20:39




$begingroup$
What do you mean by $exp$? As far as I know, it's meant to be a function, but you're not giving it an input.
$endgroup$
– Calvin Godfrey
Jan 10 at 20:39












$begingroup$
@CalvinGodfrey, they just mean $e$.
$endgroup$
– Joe
Jan 10 at 20:40




$begingroup$
@CalvinGodfrey, they just mean $e$.
$endgroup$
– Joe
Jan 10 at 20:40




3




3




$begingroup$
@Joe In that case $exp(1)$ would be more accurate.
$endgroup$
– cansomeonehelpmeout
Jan 10 at 20:41




$begingroup$
@Joe In that case $exp(1)$ would be more accurate.
$endgroup$
– cansomeonehelpmeout
Jan 10 at 20:41












$begingroup$
@cansomeonehelpmeout, yes that would have been more appropriate. I'm just explaining what OP meant. I absolutely do not condone their notation.
$endgroup$
– Joe
Jan 10 at 20:43




$begingroup$
@cansomeonehelpmeout, yes that would have been more appropriate. I'm just explaining what OP meant. I absolutely do not condone their notation.
$endgroup$
– Joe
Jan 10 at 20:43












$begingroup$
@mfl, thanks, I was stuck because I didn't consider that all 3 of expressions in this inequality are sequences and thought in terms of fixed n
$endgroup$
– dpd
Jan 10 at 21:02




$begingroup$
@mfl, thanks, I was stuck because I didn't consider that all 3 of expressions in this inequality are sequences and thought in terms of fixed n
$endgroup$
– dpd
Jan 10 at 21:02










1 Answer
1






active

oldest

votes


















4












$begingroup$

Actually yes, we can make $epsilon$ arbitrarily small. Note that your argument is a limit argument, meaning it doesn't hold for every $n$. It only holds for all $n geq N$, where $N$ depends on $epsilon$.






share|cite|improve this answer









$endgroup$














    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3069144%2ffracnen-n-fracne-n-varepsilonn-doesnt-comply-with%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4












    $begingroup$

    Actually yes, we can make $epsilon$ arbitrarily small. Note that your argument is a limit argument, meaning it doesn't hold for every $n$. It only holds for all $n geq N$, where $N$ depends on $epsilon$.






    share|cite|improve this answer









    $endgroup$


















      4












      $begingroup$

      Actually yes, we can make $epsilon$ arbitrarily small. Note that your argument is a limit argument, meaning it doesn't hold for every $n$. It only holds for all $n geq N$, where $N$ depends on $epsilon$.






      share|cite|improve this answer









      $endgroup$
















        4












        4








        4





        $begingroup$

        Actually yes, we can make $epsilon$ arbitrarily small. Note that your argument is a limit argument, meaning it doesn't hold for every $n$. It only holds for all $n geq N$, where $N$ depends on $epsilon$.






        share|cite|improve this answer









        $endgroup$



        Actually yes, we can make $epsilon$ arbitrarily small. Note that your argument is a limit argument, meaning it doesn't hold for every $n$. It only holds for all $n geq N$, where $N$ depends on $epsilon$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 10 at 20:42









        JoeJoe

        75129




        75129






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3069144%2ffracnen-n-fracne-n-varepsilonn-doesnt-comply-with%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Bressuire

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more

            Cabo Verde

            Image
              Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
            Read more

            Gyllenstierna

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more