How to evaluate $ lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n $ where $i=sqrt{-1}$?












0















How to evaluate the following limit?
$$
lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
$$
Here $i=sqrt{-1}$.




I got:
$$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
= lim_{nto infty} frac{(i-1)^n}{n(sqrt{2})^n}
$$
I know the lower part goes to infinity but what to do with the upper part? Is that usefull to use squeeze theorem or is there any simplier way?






sequences-and-series limits complex-numbers







share|cite|improve this question




















  • 1




    It's very hard to understand what you wrote. Try mathjax
    – DonAntonio
    Nov 24 at 21:38












  • Express $(1+i)/sqrt{2}=exp(ipi/4)$.
    – Diger
    Nov 24 at 21:39








  • 1




    L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as $sqrt{blah blah}$. A limit of a function f(x) from $n to infty$ can be written $lim_{nto infty} f(x)$. Just a few pointers.
    – amWhy
    Nov 24 at 21:43






  • 1




    If your function $f(x)$ is a fraction, you can write it as follows $f(x) = frac{"numerator here"}{"denominator here"}$
    – amWhy
    Nov 24 at 21:46












  • Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
    – Shubham Johri
    Nov 24 at 21:57
















0















How to evaluate the following limit?
$$
lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
$$
Here $i=sqrt{-1}$.




I got:
$$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
= lim_{nto infty} frac{(i-1)^n}{n(sqrt{2})^n}
$$
I know the lower part goes to infinity but what to do with the upper part? Is that usefull to use squeeze theorem or is there any simplier way?






sequences-and-series limits complex-numbers







share|cite|improve this question




















  • 1




    It's very hard to understand what you wrote. Try mathjax
    – DonAntonio
    Nov 24 at 21:38












  • Express $(1+i)/sqrt{2}=exp(ipi/4)$.
    – Diger
    Nov 24 at 21:39








  • 1




    L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as $sqrt{blah blah}$. A limit of a function f(x) from $n to infty$ can be written $lim_{nto infty} f(x)$. Just a few pointers.
    – amWhy
    Nov 24 at 21:43






  • 1




    If your function $f(x)$ is a fraction, you can write it as follows $f(x) = frac{"numerator here"}{"denominator here"}$
    – amWhy
    Nov 24 at 21:46












  • Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
    – Shubham Johri
    Nov 24 at 21:57














0












0








0








How to evaluate the following limit?
$$
lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
$$
Here $i=sqrt{-1}$.




I got:
$$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
= lim_{nto infty} frac{(i-1)^n}{n(sqrt{2})^n}
$$
I know the lower part goes to infinity but what to do with the upper part? Is that usefull to use squeeze theorem or is there any simplier way?






sequences-and-series limits complex-numbers







share|cite|improve this question
















How to evaluate the following limit?
$$
lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
$$
Here $i=sqrt{-1}$.




I got:
$$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n
= lim_{nto infty} frac{(i-1)^n}{n(sqrt{2})^n}
$$
I know the lower part goes to infinity but what to do with the upper part? Is that usefull to use squeeze theorem or is there any simplier way?






sequences-and-series limits complex-numbers




sequences-and-series limits complex-numbers






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 24 at 23:16









user587192

1,770213




1,770213










asked Nov 24 at 21:35









L. Spy

12




12








  • 1




    It's very hard to understand what you wrote. Try mathjax
    – DonAntonio
    Nov 24 at 21:38












  • Express $(1+i)/sqrt{2}=exp(ipi/4)$.
    – Diger
    Nov 24 at 21:39








  • 1




    L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as $sqrt{blah blah}$. A limit of a function f(x) from $n to infty$ can be written $lim_{nto infty} f(x)$. Just a few pointers.
    – amWhy
    Nov 24 at 21:43






  • 1




    If your function $f(x)$ is a fraction, you can write it as follows $f(x) = frac{"numerator here"}{"denominator here"}$
    – amWhy
    Nov 24 at 21:46












  • Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
    – Shubham Johri
    Nov 24 at 21:57














  • 1




    It's very hard to understand what you wrote. Try mathjax
    – DonAntonio
    Nov 24 at 21:38












  • Express $(1+i)/sqrt{2}=exp(ipi/4)$.
    – Diger
    Nov 24 at 21:39








  • 1




    L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as $sqrt{blah blah}$. A limit of a function f(x) from $n to infty$ can be written $lim_{nto infty} f(x)$. Just a few pointers.
    – amWhy
    Nov 24 at 21:43






  • 1




    If your function $f(x)$ is a fraction, you can write it as follows $f(x) = frac{"numerator here"}{"denominator here"}$
    – amWhy
    Nov 24 at 21:46












  • Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
    – Shubham Johri
    Nov 24 at 21:57








1




1




It's very hard to understand what you wrote. Try mathjax
– DonAntonio
Nov 24 at 21:38






It's very hard to understand what you wrote. Try mathjax
– DonAntonio
Nov 24 at 21:38














Express $(1+i)/sqrt{2}=exp(ipi/4)$.
– Diger
Nov 24 at 21:39






Express $(1+i)/sqrt{2}=exp(ipi/4)$.
– Diger
Nov 24 at 21:39






1




1




L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as $sqrt{blah blah}$. A limit of a function f(x) from $n to infty$ can be written $lim_{nto infty} f(x)$. Just a few pointers.
– amWhy
Nov 24 at 21:43




L.Spy Talk of "mathjax" might be confusing to you. Sorry about that. Mathjax is a way to format mathematical expressions so they render very nicely, like you'd see in a textbook. Here is a really handy tutorial for learning mathjax. Anything you want under a square root sign, you can format as $sqrt{blah blah}$. A limit of a function f(x) from $n to infty$ can be written $lim_{nto infty} f(x)$. Just a few pointers.
– amWhy
Nov 24 at 21:43




1




1




If your function $f(x)$ is a fraction, you can write it as follows $f(x) = frac{"numerator here"}{"denominator here"}$
– amWhy
Nov 24 at 21:46






If your function $f(x)$ is a fraction, you can write it as follows $f(x) = frac{"numerator here"}{"denominator here"}$
– amWhy
Nov 24 at 21:46














Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
– Shubham Johri
Nov 24 at 21:57




Write $frac{1+i}{sqrt{2}}=e^{ipi/4}implies (frac{1+i}{sqrt{2}})^n=e^{i npi/4}$
– Shubham Johri
Nov 24 at 21:57










2 Answers
2






active

oldest

votes


















4














We have
$$
left|frac{1+i}{sqrt{2}}right|=1
$$
Thus
$$
left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
$$






share|cite|improve this answer





















  • egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
    – L. Spy
    Nov 25 at 9:25










  • @L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
    – egreg
    Nov 25 at 9:40





















1














$$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$






share|cite|improve this answer





















    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%2f3012114%2fhow-to-evaluate-lim-n-to-infty-fracin-left-frac1i-sqrt2-right%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4














    We have
    $$
    left|frac{1+i}{sqrt{2}}right|=1
    $$
    Thus
    $$
    left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
    $$






    share|cite|improve this answer





















    • egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
      – L. Spy
      Nov 25 at 9:25










    • @L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
      – egreg
      Nov 25 at 9:40


















    4














    We have
    $$
    left|frac{1+i}{sqrt{2}}right|=1
    $$
    Thus
    $$
    left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
    $$






    share|cite|improve this answer





















    • egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
      – L. Spy
      Nov 25 at 9:25










    • @L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
      – egreg
      Nov 25 at 9:40
















    4












    4








    4






    We have
    $$
    left|frac{1+i}{sqrt{2}}right|=1
    $$
    Thus
    $$
    left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
    $$






    share|cite|improve this answer












    We have
    $$
    left|frac{1+i}{sqrt{2}}right|=1
    $$
    Thus
    $$
    left|frac{i}{n}left(frac{1+i}{sqrt{2}}right)^{!n}right|=frac{1}{n}
    $$







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Nov 24 at 22:53









    egreg

    178k1484201




    178k1484201












    • egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
      – L. Spy
      Nov 25 at 9:25










    • @L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
      – egreg
      Nov 25 at 9:40




















    • egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
      – L. Spy
      Nov 25 at 9:25










    • @L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
      – egreg
      Nov 25 at 9:40


















    egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
    – L. Spy
    Nov 25 at 9:25




    egreg-How it could be 1? I thought that (1+i)/√2 is one of the roots of i=(a+bi)^2
    – L. Spy
    Nov 25 at 9:25












    @L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
    – egreg
    Nov 25 at 9:40






    @L.Spy $sqrt{(1/sqrt{2})^2+(1/sqrt{2})^2}=sqrt{1/2+1/2}=1$. With your (more complicated) approach: since $|i|=1$, also its square roots have modulus $1$.
    – egreg
    Nov 25 at 9:40













    1














    $$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$






    share|cite|improve this answer


























      1














      $$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$






      share|cite|improve this answer
























        1












        1








        1






        $$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$






        share|cite|improve this answer












        $$lim_{nto infty} |frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n|=lim_{nto infty} {1over n}=0$$therefore $$lim_{nto infty} frac{i}{n}left(frac{1+i}{sqrt{2}}right)^n=0$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 28 at 0:31









        Mostafa Ayaz

        13.7k3836




        13.7k3836






























            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.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • 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.


            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%2f3012114%2fhow-to-evaluate-lim-n-to-infty-fracin-left-frac1i-sqrt2-right%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

            Plaza Victoria

            Image
            Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l...
            Read more

            Puebla de Zaragoza

            Image
            Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
            Read more

            Musa

            Image
            Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
            Read more