Implications of Convergence of a Series

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












Let us say that:
$$
sum_k = 1^infty fraca_kk^2 <infty
$$
Where the $a_k geq 0$. Can we say anything about what the limit:
$$
lim_nto infty sum_k = 1^nfraca_kn^2
$$
converges to? I know the limit converges by the comparison test, but is there anything we can say about its value?




real-analysis sequences-and-series limits




share|cite|improve this question

















  • 1




    A great way to find the sum of an infinite series is making a telescopic series. See if this helps.
    – Anik Bhowmick
    3 hours ago














up vote
1
down vote

favorite












Let us say that:
$$
sum_k = 1^infty fraca_kk^2 <infty
$$
Where the $a_k geq 0$. Can we say anything about what the limit:
$$
lim_nto infty sum_k = 1^nfraca_kn^2
$$
converges to? I know the limit converges by the comparison test, but is there anything we can say about its value?




real-analysis sequences-and-series limits




share|cite|improve this question

















  • 1




    A great way to find the sum of an infinite series is making a telescopic series. See if this helps.
    – Anik Bhowmick
    3 hours ago












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Let us say that:
$$
sum_k = 1^infty fraca_kk^2 <infty
$$
Where the $a_k geq 0$. Can we say anything about what the limit:
$$
lim_nto infty sum_k = 1^nfraca_kn^2
$$
converges to? I know the limit converges by the comparison test, but is there anything we can say about its value?




real-analysis sequences-and-series limits




share|cite|improve this question













Let us say that:
$$
sum_k = 1^infty fraca_kk^2 <infty
$$
Where the $a_k geq 0$. Can we say anything about what the limit:
$$
lim_nto infty sum_k = 1^nfraca_kn^2
$$
converges to? I know the limit converges by the comparison test, but is there anything we can say about its value?





real-analysis sequences-and-series limits





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 3 hours ago









TheSimpliFire

9,17651651




9,17651651









asked 3 hours ago









rubikscube09

843516




843516







  • 1




    A great way to find the sum of an infinite series is making a telescopic series. See if this helps.
    – Anik Bhowmick
    3 hours ago












  • 1




    A great way to find the sum of an infinite series is making a telescopic series. See if this helps.
    – Anik Bhowmick
    3 hours ago







1




1




A great way to find the sum of an infinite series is making a telescopic series. See if this helps.
– Anik Bhowmick
3 hours ago




A great way to find the sum of an infinite series is making a telescopic series. See if this helps.
– Anik Bhowmick
3 hours ago










2 Answers
2






active

oldest

votes

















up vote
0
down vote













Let $s_n=sum_k=1^na_k$.
By Stolz–Cesàro theorem, we have
beginalign
lim_ntoinftyfrac 1n^2sum_k=1^na_k
&=lim_ntoinftyfracs_nn^2\
&=lim_ntoinftyfracs_n+1-s_n(n+1)^2-n^2\
&=lim_ntoinftyfraca_n+12n+1\
&=frac 12lim_ntoinftyfraca_n+1n+1\
&=frac 12lim_ntoinftyfraca_n+1-a_n(n+1)-n\
&=frac 12lim_ntoinfty(a_n+1-a_n)
endalign
provided that the last limit exists.






share|cite|improve this answer























  • If $L=lim_nto infty frac a_n+1n+1$ exists then $L= 0.$ Because if $L>0$ then $a_k/k>L/2$ and hence $a_k/k^2>L/2k$ for all but finitely many $k,$ implying that $sum_ka_k/k^2$ diverges
    – DanielWainfleet
    1 hour ago


















up vote
0
down vote













Let $S_n = sum_k=1^n fraca_kk^2$ where $lim_n to inftyS_n = S$.



Summing by parts we get



$$sum_k=1^n a_k = sum_k=1^n k^2 fraca_kk^2 = n^2S_n + sum_k=1^n-1S_k left(k^2 - (k+1)^2 right),$$



and



$$frac1n^2sum_k=1^n a_k = S_n - frac1n^2sum_k=1^n-1(2k+1)S_k$$



The first term on the RHS converges to $S$ and the second term on the RHS converges to $S$ by Stolz-Cesaro:



$$lim_n to inftyfrac1n^2sum_k=1^n-1(2k+1)S_k = lim_n to inftyfrac(2n+1)S_n(n+1)^2 - n^2 = lim_n to inftyS_n = S$$



Hence,



$$lim_n to infty frac1n^2sum_k=1^n a_k = 0$$






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',
    convertImagesToLinks: true,
    noModals: false,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    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%2f2873259%2fimplications-of-convergence-of-a-series%23new-answer', 'question_page');

    );

    Post as a guest

















    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    0
    down vote













    Let $s_n=sum_k=1^na_k$.
    By Stolz–Cesàro theorem, we have
    beginalign
    lim_ntoinftyfrac 1n^2sum_k=1^na_k
    &=lim_ntoinftyfracs_nn^2\
    &=lim_ntoinftyfracs_n+1-s_n(n+1)^2-n^2\
    &=lim_ntoinftyfraca_n+12n+1\
    &=frac 12lim_ntoinftyfraca_n+1n+1\
    &=frac 12lim_ntoinftyfraca_n+1-a_n(n+1)-n\
    &=frac 12lim_ntoinfty(a_n+1-a_n)
    endalign
    provided that the last limit exists.






    share|cite|improve this answer























    • If $L=lim_nto infty frac a_n+1n+1$ exists then $L= 0.$ Because if $L>0$ then $a_k/k>L/2$ and hence $a_k/k^2>L/2k$ for all but finitely many $k,$ implying that $sum_ka_k/k^2$ diverges
      – DanielWainfleet
      1 hour ago















    up vote
    0
    down vote













    Let $s_n=sum_k=1^na_k$.
    By Stolz–Cesàro theorem, we have
    beginalign
    lim_ntoinftyfrac 1n^2sum_k=1^na_k
    &=lim_ntoinftyfracs_nn^2\
    &=lim_ntoinftyfracs_n+1-s_n(n+1)^2-n^2\
    &=lim_ntoinftyfraca_n+12n+1\
    &=frac 12lim_ntoinftyfraca_n+1n+1\
    &=frac 12lim_ntoinftyfraca_n+1-a_n(n+1)-n\
    &=frac 12lim_ntoinfty(a_n+1-a_n)
    endalign
    provided that the last limit exists.






    share|cite|improve this answer























    • If $L=lim_nto infty frac a_n+1n+1$ exists then $L= 0.$ Because if $L>0$ then $a_k/k>L/2$ and hence $a_k/k^2>L/2k$ for all but finitely many $k,$ implying that $sum_ka_k/k^2$ diverges
      – DanielWainfleet
      1 hour ago













    up vote
    0
    down vote










    up vote
    0
    down vote









    Let $s_n=sum_k=1^na_k$.
    By Stolz–Cesàro theorem, we have
    beginalign
    lim_ntoinftyfrac 1n^2sum_k=1^na_k
    &=lim_ntoinftyfracs_nn^2\
    &=lim_ntoinftyfracs_n+1-s_n(n+1)^2-n^2\
    &=lim_ntoinftyfraca_n+12n+1\
    &=frac 12lim_ntoinftyfraca_n+1n+1\
    &=frac 12lim_ntoinftyfraca_n+1-a_n(n+1)-n\
    &=frac 12lim_ntoinfty(a_n+1-a_n)
    endalign
    provided that the last limit exists.






    share|cite|improve this answer















    Let $s_n=sum_k=1^na_k$.
    By Stolz–Cesàro theorem, we have
    beginalign
    lim_ntoinftyfrac 1n^2sum_k=1^na_k
    &=lim_ntoinftyfracs_nn^2\
    &=lim_ntoinftyfracs_n+1-s_n(n+1)^2-n^2\
    &=lim_ntoinftyfraca_n+12n+1\
    &=frac 12lim_ntoinftyfraca_n+1n+1\
    &=frac 12lim_ntoinftyfraca_n+1-a_n(n+1)-n\
    &=frac 12lim_ntoinfty(a_n+1-a_n)
    endalign
    provided that the last limit exists.







    share|cite|improve this answer















    share|cite|improve this answer



    share|cite|improve this answer








    edited 2 hours ago


























    answered 3 hours ago









    Fabio Lucchini

    5,55411025




    5,55411025











    • If $L=lim_nto infty frac a_n+1n+1$ exists then $L= 0.$ Because if $L>0$ then $a_k/k>L/2$ and hence $a_k/k^2>L/2k$ for all but finitely many $k,$ implying that $sum_ka_k/k^2$ diverges
      – DanielWainfleet
      1 hour ago

















    • If $L=lim_nto infty frac a_n+1n+1$ exists then $L= 0.$ Because if $L>0$ then $a_k/k>L/2$ and hence $a_k/k^2>L/2k$ for all but finitely many $k,$ implying that $sum_ka_k/k^2$ diverges
      – DanielWainfleet
      1 hour ago
















    If $L=lim_nto infty frac a_n+1n+1$ exists then $L= 0.$ Because if $L>0$ then $a_k/k>L/2$ and hence $a_k/k^2>L/2k$ for all but finitely many $k,$ implying that $sum_ka_k/k^2$ diverges
    – DanielWainfleet
    1 hour ago





    If $L=lim_nto infty frac a_n+1n+1$ exists then $L= 0.$ Because if $L>0$ then $a_k/k>L/2$ and hence $a_k/k^2>L/2k$ for all but finitely many $k,$ implying that $sum_ka_k/k^2$ diverges
    – DanielWainfleet
    1 hour ago











    up vote
    0
    down vote













    Let $S_n = sum_k=1^n fraca_kk^2$ where $lim_n to inftyS_n = S$.



    Summing by parts we get



    $$sum_k=1^n a_k = sum_k=1^n k^2 fraca_kk^2 = n^2S_n + sum_k=1^n-1S_k left(k^2 - (k+1)^2 right),$$



    and



    $$frac1n^2sum_k=1^n a_k = S_n - frac1n^2sum_k=1^n-1(2k+1)S_k$$



    The first term on the RHS converges to $S$ and the second term on the RHS converges to $S$ by Stolz-Cesaro:



    $$lim_n to inftyfrac1n^2sum_k=1^n-1(2k+1)S_k = lim_n to inftyfrac(2n+1)S_n(n+1)^2 - n^2 = lim_n to inftyS_n = S$$



    Hence,



    $$lim_n to infty frac1n^2sum_k=1^n a_k = 0$$






    share|cite|improve this answer



























      up vote
      0
      down vote













      Let $S_n = sum_k=1^n fraca_kk^2$ where $lim_n to inftyS_n = S$.



      Summing by parts we get



      $$sum_k=1^n a_k = sum_k=1^n k^2 fraca_kk^2 = n^2S_n + sum_k=1^n-1S_k left(k^2 - (k+1)^2 right),$$



      and



      $$frac1n^2sum_k=1^n a_k = S_n - frac1n^2sum_k=1^n-1(2k+1)S_k$$



      The first term on the RHS converges to $S$ and the second term on the RHS converges to $S$ by Stolz-Cesaro:



      $$lim_n to inftyfrac1n^2sum_k=1^n-1(2k+1)S_k = lim_n to inftyfrac(2n+1)S_n(n+1)^2 - n^2 = lim_n to inftyS_n = S$$



      Hence,



      $$lim_n to infty frac1n^2sum_k=1^n a_k = 0$$






      share|cite|improve this answer

























        up vote
        0
        down vote










        up vote
        0
        down vote









        Let $S_n = sum_k=1^n fraca_kk^2$ where $lim_n to inftyS_n = S$.



        Summing by parts we get



        $$sum_k=1^n a_k = sum_k=1^n k^2 fraca_kk^2 = n^2S_n + sum_k=1^n-1S_k left(k^2 - (k+1)^2 right),$$



        and



        $$frac1n^2sum_k=1^n a_k = S_n - frac1n^2sum_k=1^n-1(2k+1)S_k$$



        The first term on the RHS converges to $S$ and the second term on the RHS converges to $S$ by Stolz-Cesaro:



        $$lim_n to inftyfrac1n^2sum_k=1^n-1(2k+1)S_k = lim_n to inftyfrac(2n+1)S_n(n+1)^2 - n^2 = lim_n to inftyS_n = S$$



        Hence,



        $$lim_n to infty frac1n^2sum_k=1^n a_k = 0$$






        share|cite|improve this answer















        Let $S_n = sum_k=1^n fraca_kk^2$ where $lim_n to inftyS_n = S$.



        Summing by parts we get



        $$sum_k=1^n a_k = sum_k=1^n k^2 fraca_kk^2 = n^2S_n + sum_k=1^n-1S_k left(k^2 - (k+1)^2 right),$$



        and



        $$frac1n^2sum_k=1^n a_k = S_n - frac1n^2sum_k=1^n-1(2k+1)S_k$$



        The first term on the RHS converges to $S$ and the second term on the RHS converges to $S$ by Stolz-Cesaro:



        $$lim_n to inftyfrac1n^2sum_k=1^n-1(2k+1)S_k = lim_n to inftyfrac(2n+1)S_n(n+1)^2 - n^2 = lim_n to inftyS_n = S$$



        Hence,



        $$lim_n to infty frac1n^2sum_k=1^n a_k = 0$$







        share|cite|improve this answer















        share|cite|improve this answer



        share|cite|improve this answer








        edited 47 mins ago


























        answered 1 hour ago









        RRL

        43.3k42160




        43.3k42160






















             

            draft saved


            draft discarded


























             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2873259%2fimplications-of-convergence-of-a-series%23new-answer', 'question_page');

            );

            Post as a guest
































































            Comments

            Post a Comment

            Popular posts from this blog

            What is the equation of a 3D cone with generalised tilt?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
            Read more

            Color the edges and diagonals of a regular polygon

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
            Read more

            Relationship between determinant of matrix and determinant of adjoint?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
            Read more