If $sum a_k$ converges, does $sum(a_{k+1}- 2 a_{k+3})$ converge as well?












6












$begingroup$



If $sum_{k=0}^infty a_k$ is convergent with value $s$, what about $sum_{k=0}^infty b_k$ where $b_k=a_{k+1}- 2 a_{k+3}$?




My reasoning:
$$sum_{k=0}^infty b_k =lim_{n rightarrow infty} sum_{k=0}^n b_k=lim_{n rightarrow infty} sum_{k=0}^na_{k+1}-2a_{k+3}
$$

Within the sum we are only dealing with finitely many terms, we can split up the sum and take the limit afterwards:
$$ lim_{n rightarrow infty} sum_{k=0}^n b_k=lim_{n rightarrow infty} left( sum_{k=0}^n a_{k+1}- sum_{k=0}^n 2a_{k+3} right) $$
Now we want to get $s$ in here, the value of our sum, we need to do some index juggling, since our sum is not of the right form yet.
$$lim_{n rightarrow infty} left( sum_{k=0}^n a_{k} - a_0- 2sum_{k=0}^n a_{k} +2a_0 + 2a_1+2a_2right) $$
Here we applied an index shift, but then we need to compensate for the terms that we added to the sum, we finally apply the limit and get:
$$sum_{k=0}^infty b_k=s +a_0-2s+2a_1+2a_2= a_0+2a_1+2a_2 -s $$



Did that all make sense, is my reasoning correct?
conclusion: it converges as we computed the exact value.










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    If $sum x_n = X in mathbb{R}$ and $sum y_n = Y in mathbb{R}$, then $sum (x_n + y_n) =X+ Y$, so yes, the reasoning is correct.
    $endgroup$
    – Cosmin
    Dec 2 '18 at 13:40








  • 1




    $begingroup$
    Looks fine to me
    $endgroup$
    – Shubham Johri
    Dec 2 '18 at 14:22












  • $begingroup$
    Please provide your answers in the answer section. This way the question does not stay open.
    $endgroup$
    – Wesley Strik
    Dec 3 '18 at 11:31










  • $begingroup$
    math.meta.stackexchange.com/questions/1559/…
    $endgroup$
    – Wesley Strik
    Dec 3 '18 at 11:33
















6












$begingroup$



If $sum_{k=0}^infty a_k$ is convergent with value $s$, what about $sum_{k=0}^infty b_k$ where $b_k=a_{k+1}- 2 a_{k+3}$?




My reasoning:
$$sum_{k=0}^infty b_k =lim_{n rightarrow infty} sum_{k=0}^n b_k=lim_{n rightarrow infty} sum_{k=0}^na_{k+1}-2a_{k+3}
$$

Within the sum we are only dealing with finitely many terms, we can split up the sum and take the limit afterwards:
$$ lim_{n rightarrow infty} sum_{k=0}^n b_k=lim_{n rightarrow infty} left( sum_{k=0}^n a_{k+1}- sum_{k=0}^n 2a_{k+3} right) $$
Now we want to get $s$ in here, the value of our sum, we need to do some index juggling, since our sum is not of the right form yet.
$$lim_{n rightarrow infty} left( sum_{k=0}^n a_{k} - a_0- 2sum_{k=0}^n a_{k} +2a_0 + 2a_1+2a_2right) $$
Here we applied an index shift, but then we need to compensate for the terms that we added to the sum, we finally apply the limit and get:
$$sum_{k=0}^infty b_k=s +a_0-2s+2a_1+2a_2= a_0+2a_1+2a_2 -s $$



Did that all make sense, is my reasoning correct?
conclusion: it converges as we computed the exact value.










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    If $sum x_n = X in mathbb{R}$ and $sum y_n = Y in mathbb{R}$, then $sum (x_n + y_n) =X+ Y$, so yes, the reasoning is correct.
    $endgroup$
    – Cosmin
    Dec 2 '18 at 13:40








  • 1




    $begingroup$
    Looks fine to me
    $endgroup$
    – Shubham Johri
    Dec 2 '18 at 14:22












  • $begingroup$
    Please provide your answers in the answer section. This way the question does not stay open.
    $endgroup$
    – Wesley Strik
    Dec 3 '18 at 11:31










  • $begingroup$
    math.meta.stackexchange.com/questions/1559/…
    $endgroup$
    – Wesley Strik
    Dec 3 '18 at 11:33














6












6








6





$begingroup$



If $sum_{k=0}^infty a_k$ is convergent with value $s$, what about $sum_{k=0}^infty b_k$ where $b_k=a_{k+1}- 2 a_{k+3}$?




My reasoning:
$$sum_{k=0}^infty b_k =lim_{n rightarrow infty} sum_{k=0}^n b_k=lim_{n rightarrow infty} sum_{k=0}^na_{k+1}-2a_{k+3}
$$

Within the sum we are only dealing with finitely many terms, we can split up the sum and take the limit afterwards:
$$ lim_{n rightarrow infty} sum_{k=0}^n b_k=lim_{n rightarrow infty} left( sum_{k=0}^n a_{k+1}- sum_{k=0}^n 2a_{k+3} right) $$
Now we want to get $s$ in here, the value of our sum, we need to do some index juggling, since our sum is not of the right form yet.
$$lim_{n rightarrow infty} left( sum_{k=0}^n a_{k} - a_0- 2sum_{k=0}^n a_{k} +2a_0 + 2a_1+2a_2right) $$
Here we applied an index shift, but then we need to compensate for the terms that we added to the sum, we finally apply the limit and get:
$$sum_{k=0}^infty b_k=s +a_0-2s+2a_1+2a_2= a_0+2a_1+2a_2 -s $$



Did that all make sense, is my reasoning correct?
conclusion: it converges as we computed the exact value.










share|cite|improve this question











$endgroup$





If $sum_{k=0}^infty a_k$ is convergent with value $s$, what about $sum_{k=0}^infty b_k$ where $b_k=a_{k+1}- 2 a_{k+3}$?




My reasoning:
$$sum_{k=0}^infty b_k =lim_{n rightarrow infty} sum_{k=0}^n b_k=lim_{n rightarrow infty} sum_{k=0}^na_{k+1}-2a_{k+3}
$$

Within the sum we are only dealing with finitely many terms, we can split up the sum and take the limit afterwards:
$$ lim_{n rightarrow infty} sum_{k=0}^n b_k=lim_{n rightarrow infty} left( sum_{k=0}^n a_{k+1}- sum_{k=0}^n 2a_{k+3} right) $$
Now we want to get $s$ in here, the value of our sum, we need to do some index juggling, since our sum is not of the right form yet.
$$lim_{n rightarrow infty} left( sum_{k=0}^n a_{k} - a_0- 2sum_{k=0}^n a_{k} +2a_0 + 2a_1+2a_2right) $$
Here we applied an index shift, but then we need to compensate for the terms that we added to the sum, we finally apply the limit and get:
$$sum_{k=0}^infty b_k=s +a_0-2s+2a_1+2a_2= a_0+2a_1+2a_2 -s $$



Did that all make sense, is my reasoning correct?
conclusion: it converges as we computed the exact value.







real-analysis sequences-and-series proof-verification






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 3 '18 at 11:32









Did

247k23222458




247k23222458










asked Dec 2 '18 at 13:07









Wesley StrikWesley Strik

1,653423




1,653423








  • 2




    $begingroup$
    If $sum x_n = X in mathbb{R}$ and $sum y_n = Y in mathbb{R}$, then $sum (x_n + y_n) =X+ Y$, so yes, the reasoning is correct.
    $endgroup$
    – Cosmin
    Dec 2 '18 at 13:40








  • 1




    $begingroup$
    Looks fine to me
    $endgroup$
    – Shubham Johri
    Dec 2 '18 at 14:22












  • $begingroup$
    Please provide your answers in the answer section. This way the question does not stay open.
    $endgroup$
    – Wesley Strik
    Dec 3 '18 at 11:31










  • $begingroup$
    math.meta.stackexchange.com/questions/1559/…
    $endgroup$
    – Wesley Strik
    Dec 3 '18 at 11:33














  • 2




    $begingroup$
    If $sum x_n = X in mathbb{R}$ and $sum y_n = Y in mathbb{R}$, then $sum (x_n + y_n) =X+ Y$, so yes, the reasoning is correct.
    $endgroup$
    – Cosmin
    Dec 2 '18 at 13:40








  • 1




    $begingroup$
    Looks fine to me
    $endgroup$
    – Shubham Johri
    Dec 2 '18 at 14:22












  • $begingroup$
    Please provide your answers in the answer section. This way the question does not stay open.
    $endgroup$
    – Wesley Strik
    Dec 3 '18 at 11:31










  • $begingroup$
    math.meta.stackexchange.com/questions/1559/…
    $endgroup$
    – Wesley Strik
    Dec 3 '18 at 11:33








2




2




$begingroup$
If $sum x_n = X in mathbb{R}$ and $sum y_n = Y in mathbb{R}$, then $sum (x_n + y_n) =X+ Y$, so yes, the reasoning is correct.
$endgroup$
– Cosmin
Dec 2 '18 at 13:40






$begingroup$
If $sum x_n = X in mathbb{R}$ and $sum y_n = Y in mathbb{R}$, then $sum (x_n + y_n) =X+ Y$, so yes, the reasoning is correct.
$endgroup$
– Cosmin
Dec 2 '18 at 13:40






1




1




$begingroup$
Looks fine to me
$endgroup$
– Shubham Johri
Dec 2 '18 at 14:22






$begingroup$
Looks fine to me
$endgroup$
– Shubham Johri
Dec 2 '18 at 14:22














$begingroup$
Please provide your answers in the answer section. This way the question does not stay open.
$endgroup$
– Wesley Strik
Dec 3 '18 at 11:31




$begingroup$
Please provide your answers in the answer section. This way the question does not stay open.
$endgroup$
– Wesley Strik
Dec 3 '18 at 11:31












$begingroup$
math.meta.stackexchange.com/questions/1559/…
$endgroup$
– Wesley Strik
Dec 3 '18 at 11:33




$begingroup$
math.meta.stackexchange.com/questions/1559/…
$endgroup$
– Wesley Strik
Dec 3 '18 at 11:33










1 Answer
1






active

oldest

votes


















1












$begingroup$

I will write this in the answer section so that the question does not remain open.



The reasoning looks fine and correct to me.






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%2f3022620%2fif-sum-a-k-converges-does-suma-k1-2-a-k3-converge-as-well%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









    1












    $begingroup$

    I will write this in the answer section so that the question does not remain open.



    The reasoning looks fine and correct to me.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      I will write this in the answer section so that the question does not remain open.



      The reasoning looks fine and correct to me.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        I will write this in the answer section so that the question does not remain open.



        The reasoning looks fine and correct to me.






        share|cite|improve this answer









        $endgroup$



        I will write this in the answer section so that the question does not remain open.



        The reasoning looks fine and correct to me.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 4 '18 at 17:55









        CosminCosmin

        1,4241524




        1,4241524






























            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%2f3022620%2fif-sum-a-k-converges-does-suma-k1-2-a-k3-converge-as-well%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

            Ellipse (mathématiques)

            Quarter-circle Tiles

            Mont Emei