Computing a series sumHow to prove $sum_n=0^infty fracn^22^n = 6$?Writing an infinite series as the sum of the seriesComputing the sum of an infinite seriescomputing the series $sum_n=1^infty frac1n^2 2^n$Sum of a series to an exact answerConfused computing sum of Fourier seriesClosed form of this series?Prove that the given series is convergent.Sum of reciprocals of the square roots of the first N Natural NumbersSum function for power seriesShowing the sum of a power series is less than P$x$

Could a space colony 1g from the sun work?

Wireless headphones interfere with Wi-Fi signal on laptop

How did the horses get to space?

Break long word (not long text!) in longtable cell

Given 0s on Assignments with suspected and dismissed cheating?

Testing if os.path.exists with ArcPy?

Network latencies between opposite ends of the Earth

What is the effect of the Feeblemind spell on Ability Score Improvements?

tikzcd diagram within an array

What would a Dragon have to exhale to cause rain?

Six inch, clear plastic binary counting puzzle

How to redirect stdout to a file, and stdout+stderr to another one?

My bread in my bread maker rises and then falls down just after cooking starts

How could it be that 80% of townspeople were farmers during the Edo period in Japan?

Assembly writer vs compiler in VLIW architecture

Is the seat-belt sign activation when a pilot goes to the lavatory standard procedure?

I recently started my machine learning PhD and I have absolutely no idea what I'm doing

Why can't I share a one use code with anyone else?

Plucker relations in orthogonal Grassmannain

Developers demotivated due to working on same project for more than 2 years

What was Varys trying to do at the beginning of S08E05?

Should I communicate in my applications that I'm unemployed out of choice rather than because nobody will have me?

Were any toxic metals used in the International Space Station?

Would life always name the light from their sun "white"



Computing a series sum


How to prove $sum_n=0^infty fracn^22^n = 6$?Writing an infinite series as the sum of the seriesComputing the sum of an infinite seriescomputing the series $sum_n=1^infty frac1n^2 2^n$Sum of a series to an exact answerConfused computing sum of Fourier seriesClosed form of this series?Prove that the given series is convergent.Sum of reciprocals of the square roots of the first N Natural NumbersSum function for power seriesShowing the sum of a power series is less than P$x$













3












$begingroup$


$$sum_k=0^infty frack^23^k$$



I tried with that 2 method but I couldn't get the $n^2$ term.






sequences-and-series power-series







share|cite|improve this question









New contributor



Erinç Emre Çelikten is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$







  • 3




    $begingroup$
    You'll need to take two derivatives here and also make use of the first derivative case since taking two derivatives gives you $k(k-1) = k^2 - k$.
    $endgroup$
    – Cameron Williams
    7 hours ago











  • $begingroup$
    math.stackexchange.com/questions/593996/…
    $endgroup$
    – lab bhattacharjee
    4 hours ago










  • $begingroup$
    Possible duplicate of How to prove $sum_n=0^infty fracn^22^n = 6$?
    $endgroup$
    – Hans Lundmark
    46 mins ago















3












$begingroup$


$$sum_k=0^infty frack^23^k$$



I tried with that 2 method but I couldn't get the $n^2$ term.






sequences-and-series power-series







share|cite|improve this question









New contributor



Erinç Emre Çelikten is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$







  • 3




    $begingroup$
    You'll need to take two derivatives here and also make use of the first derivative case since taking two derivatives gives you $k(k-1) = k^2 - k$.
    $endgroup$
    – Cameron Williams
    7 hours ago











  • $begingroup$
    math.stackexchange.com/questions/593996/…
    $endgroup$
    – lab bhattacharjee
    4 hours ago










  • $begingroup$
    Possible duplicate of How to prove $sum_n=0^infty fracn^22^n = 6$?
    $endgroup$
    – Hans Lundmark
    46 mins ago













3












3








3





$begingroup$


$$sum_k=0^infty frack^23^k$$



I tried with that 2 method but I couldn't get the $n^2$ term.






sequences-and-series power-series







share|cite|improve this question









New contributor



Erinç Emre Çelikten is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$




$$sum_k=0^infty frack^23^k$$



I tried with that 2 method but I couldn't get the $n^2$ term.






sequences-and-series power-series




sequences-and-series power-series






share|cite|improve this question









New contributor



Erinç Emre Çelikten is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.










share|cite|improve this question









New contributor



Erinç Emre Çelikten is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








share|cite|improve this question




share|cite|improve this question








edited 7 hours ago









David G. Stork

12.5k41836




12.5k41836






New contributor



Erinç Emre Çelikten is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








asked 7 hours ago









Erinç Emre ÇeliktenErinç Emre Çelikten

182




182




New contributor



Erinç Emre Çelikten is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




New contributor




Erinç Emre Çelikten is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









  • 3




    $begingroup$
    You'll need to take two derivatives here and also make use of the first derivative case since taking two derivatives gives you $k(k-1) = k^2 - k$.
    $endgroup$
    – Cameron Williams
    7 hours ago











  • $begingroup$
    math.stackexchange.com/questions/593996/…
    $endgroup$
    – lab bhattacharjee
    4 hours ago










  • $begingroup$
    Possible duplicate of How to prove $sum_n=0^infty fracn^22^n = 6$?
    $endgroup$
    – Hans Lundmark
    46 mins ago












  • 3




    $begingroup$
    You'll need to take two derivatives here and also make use of the first derivative case since taking two derivatives gives you $k(k-1) = k^2 - k$.
    $endgroup$
    – Cameron Williams
    7 hours ago











  • $begingroup$
    math.stackexchange.com/questions/593996/…
    $endgroup$
    – lab bhattacharjee
    4 hours ago










  • $begingroup$
    Possible duplicate of How to prove $sum_n=0^infty fracn^22^n = 6$?
    $endgroup$
    – Hans Lundmark
    46 mins ago







3




3




$begingroup$
You'll need to take two derivatives here and also make use of the first derivative case since taking two derivatives gives you $k(k-1) = k^2 - k$.
$endgroup$
– Cameron Williams
7 hours ago





$begingroup$
You'll need to take two derivatives here and also make use of the first derivative case since taking two derivatives gives you $k(k-1) = k^2 - k$.
$endgroup$
– Cameron Williams
7 hours ago













$begingroup$
math.stackexchange.com/questions/593996/…
$endgroup$
– lab bhattacharjee
4 hours ago




$begingroup$
math.stackexchange.com/questions/593996/…
$endgroup$
– lab bhattacharjee
4 hours ago












$begingroup$
Possible duplicate of How to prove $sum_n=0^infty fracn^22^n = 6$?
$endgroup$
– Hans Lundmark
46 mins ago




$begingroup$
Possible duplicate of How to prove $sum_n=0^infty fracn^22^n = 6$?
$endgroup$
– Hans Lundmark
46 mins ago










3 Answers
3






active

oldest

votes


















5












$begingroup$

$$frac11-x=sum_k=0^inftyx^k$$
$$frac1(1-x)^2=sum_k=1^inftykx^k-1$$
Multiply by $x$
$$fracx(1-x)^2=sum_k=1^inftykx^k$$
$$left(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k-1$$
Multiply by $x$
$$xleft(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k$$
then let $x=frac13$






share|cite|improve this answer











$endgroup$




















    1












    $begingroup$

    Hint try to show
    begineqnarray*
    sum_k=0^infty k^2 x^k =fracx(1+4x+x^2)(1-x)^3.
    endeqnarray*






    share|cite|improve this answer









    $endgroup$




















      0












      $begingroup$

      There is another way to deal with this problem.



      Denote $S_n=sum_k=0^n frack^23^k$ then$frac13S_n=sum_k=1^n+1 frac(k-1)^23^k$ so $frac23S_n=sum_k=1^nfrac2k-13^k-fracn^23^n+1$.



      In this way we change the numerator from twice power of $k$ to the lower power. Using the same operation you can change the numerator to numbers without the appearance of $k$ then you can use the summation formula for geometric series.






      share|cite|improve this answer









      $endgroup$













        Your Answer








        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
        );



        );






        Erinç Emre Çelikten is a new contributor. Be nice, and check out our Code of Conduct.









        draft saved

        draft discarded


















        StackExchange.ready(
        function ()
        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3226310%2fcomputing-a-series-sum%23new-answer', 'question_page');

        );

        Post as a guest















        Required, but never shown

























        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        5












        $begingroup$

        $$frac11-x=sum_k=0^inftyx^k$$
        $$frac1(1-x)^2=sum_k=1^inftykx^k-1$$
        Multiply by $x$
        $$fracx(1-x)^2=sum_k=1^inftykx^k$$
        $$left(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k-1$$
        Multiply by $x$
        $$xleft(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k$$
        then let $x=frac13$






        share|cite|improve this answer











        $endgroup$

















          5












          $begingroup$

          $$frac11-x=sum_k=0^inftyx^k$$
          $$frac1(1-x)^2=sum_k=1^inftykx^k-1$$
          Multiply by $x$
          $$fracx(1-x)^2=sum_k=1^inftykx^k$$
          $$left(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k-1$$
          Multiply by $x$
          $$xleft(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k$$
          then let $x=frac13$






          share|cite|improve this answer











          $endgroup$















            5












            5








            5





            $begingroup$

            $$frac11-x=sum_k=0^inftyx^k$$
            $$frac1(1-x)^2=sum_k=1^inftykx^k-1$$
            Multiply by $x$
            $$fracx(1-x)^2=sum_k=1^inftykx^k$$
            $$left(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k-1$$
            Multiply by $x$
            $$xleft(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k$$
            then let $x=frac13$






            share|cite|improve this answer











            $endgroup$



            $$frac11-x=sum_k=0^inftyx^k$$
            $$frac1(1-x)^2=sum_k=1^inftykx^k-1$$
            Multiply by $x$
            $$fracx(1-x)^2=sum_k=1^inftykx^k$$
            $$left(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k-1$$
            Multiply by $x$
            $$xleft(fracx(1-x)^2right)'=sum_k=1^inftyk^2x^k$$
            then let $x=frac13$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 7 hours ago









            clathratus

            5,6861441




            5,6861441










            answered 7 hours ago









            E.H.EE.H.E

            17.5k11969




            17.5k11969





















                1












                $begingroup$

                Hint try to show
                begineqnarray*
                sum_k=0^infty k^2 x^k =fracx(1+4x+x^2)(1-x)^3.
                endeqnarray*






                share|cite|improve this answer









                $endgroup$

















                  1












                  $begingroup$

                  Hint try to show
                  begineqnarray*
                  sum_k=0^infty k^2 x^k =fracx(1+4x+x^2)(1-x)^3.
                  endeqnarray*






                  share|cite|improve this answer









                  $endgroup$















                    1












                    1








                    1





                    $begingroup$

                    Hint try to show
                    begineqnarray*
                    sum_k=0^infty k^2 x^k =fracx(1+4x+x^2)(1-x)^3.
                    endeqnarray*






                    share|cite|improve this answer









                    $endgroup$



                    Hint try to show
                    begineqnarray*
                    sum_k=0^infty k^2 x^k =fracx(1+4x+x^2)(1-x)^3.
                    endeqnarray*







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 7 hours ago









                    Donald SplutterwitDonald Splutterwit

                    23.4k21448




                    23.4k21448





















                        0












                        $begingroup$

                        There is another way to deal with this problem.



                        Denote $S_n=sum_k=0^n frack^23^k$ then$frac13S_n=sum_k=1^n+1 frac(k-1)^23^k$ so $frac23S_n=sum_k=1^nfrac2k-13^k-fracn^23^n+1$.



                        In this way we change the numerator from twice power of $k$ to the lower power. Using the same operation you can change the numerator to numbers without the appearance of $k$ then you can use the summation formula for geometric series.






                        share|cite|improve this answer









                        $endgroup$

















                          0












                          $begingroup$

                          There is another way to deal with this problem.



                          Denote $S_n=sum_k=0^n frack^23^k$ then$frac13S_n=sum_k=1^n+1 frac(k-1)^23^k$ so $frac23S_n=sum_k=1^nfrac2k-13^k-fracn^23^n+1$.



                          In this way we change the numerator from twice power of $k$ to the lower power. Using the same operation you can change the numerator to numbers without the appearance of $k$ then you can use the summation formula for geometric series.






                          share|cite|improve this answer









                          $endgroup$















                            0












                            0








                            0





                            $begingroup$

                            There is another way to deal with this problem.



                            Denote $S_n=sum_k=0^n frack^23^k$ then$frac13S_n=sum_k=1^n+1 frac(k-1)^23^k$ so $frac23S_n=sum_k=1^nfrac2k-13^k-fracn^23^n+1$.



                            In this way we change the numerator from twice power of $k$ to the lower power. Using the same operation you can change the numerator to numbers without the appearance of $k$ then you can use the summation formula for geometric series.






                            share|cite|improve this answer









                            $endgroup$



                            There is another way to deal with this problem.



                            Denote $S_n=sum_k=0^n frack^23^k$ then$frac13S_n=sum_k=1^n+1 frac(k-1)^23^k$ so $frac23S_n=sum_k=1^nfrac2k-13^k-fracn^23^n+1$.



                            In this way we change the numerator from twice power of $k$ to the lower power. Using the same operation you can change the numerator to numbers without the appearance of $k$ then you can use the summation formula for geometric series.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 7 hours ago









                            Feng ShaoFeng Shao

                            18010




                            18010




















                                Erinç Emre Çelikten is a new contributor. Be nice, and check out our Code of Conduct.









                                draft saved

                                draft discarded


















                                Erinç Emre Çelikten is a new contributor. Be nice, and check out our Code of Conduct.












                                Erinç Emre Çelikten is a new contributor. Be nice, and check out our Code of Conduct.











                                Erinç Emre Çelikten is a new contributor. Be nice, and check out our Code of Conduct.














                                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%2f3226310%2fcomputing-a-series-sum%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

                                Canceling a color specificationRandomly assigning color to Graphics3D objects?Default color for Filling in Mathematica 9Coloring specific elements of sets with a prime modified order in an array plotHow to pick a color differing significantly from the colors already in a given color list?Detection of the text colorColor numbers based on their valueCan color schemes for use with ColorData include opacity specification?My dynamic color schemes

                                Image
                                Would it be easier to colonise a living world or a dead world? Postal services in Italy - Poste Italiane vs. Friendpost vs. GPS What is /dev/null and why can't I use hx on it? Why are engines with carburetors hard to start in cold weather? How to make a gift without seeming creepy? Without exposing his identity, did Roy help his parents with money so that they can afford to stay in their home? Does the Creighton Method of Natural Family Planning have a failure rate of 3.2% or less? How do I break the broom in Untitled Goose Game? Had there been instances of national states banning harmful imports before the mid-19th C Opium Wars? D&D Monsters and Copyright Is sleeping on the groud in cold weather better than on an air mattress? Transiting through Switzerland by coach with lots of cash quadratic equations on 2 by 2 matrices Always Lubricate Skewers? I pay for a service, but I miss the broadcast Iron-age tools, is there a way to extract heavy metals ...
                                Read more

                                Invision Community Contents History See also References External links Navigation menuProprietaryinvisioncommunity.comIPS Community ForumsIPS Community Forumsthis blog entry"License Changes, IP.Board 3.4, and the Future""Interview -- Matt Mecham of Ibforums""CEO Invision Power Board, Matt Mecham Is a Liar, Thief!"IPB License Explanation 1.3, 1.3.1, 2.0, and 2.1ArchivedSecurity Fixes, Updates And Enhancements For IPB 1.3.1Archived"New Demo Accounts - Invision Power Services"the original"New Default Skin"the original"Invision Power Board 3.0.0 and Applications Released"the original"Archived copy"the original"Perpetual licenses being done away with""Release Notes - Invision Power Services""Introducing: IPS Community Suite 4!"Invision Community Release Notes

                                Image
                                Internet forum softwarePHP software Internet communityPHPMySQLdatabase management systemInternet forumCharles WarnerMatt MechamJarvis Entertainment GroupIkonboardInvisionFreeAjaxIPS Community ForumsIPS Community Forumsthis blog entry Invision Community From Wikipedia, the free encyclopedia Jump to navigation Jump to search Invision Community Developer(s) Invision Power Services Stable release 4.4.6 / August 19, 2019 ; 34 days ago  ( 2019-08-19 ) Platform PHP / MySQL Type Forum software License Proprietary Website invisioncommunity.com Invision Community (previously known as IPS Community Suite ) is primarily an Internet community software produced by Invision Power Services, Inc. It is written in PHP and uses MySQL as a database management system. Invision Power Services sell applications that each can be bought and installed separately in addition to the Suite, the most widely known being the Internet forum software Invision Power Boar...
                                Read more

                                199年 目錄 大件事 到箇年出世嗰人 到箇年死嗰人 節慶、風俗習慣 導覽選單

                                Image
                                199年 平年 199年 出自維基百科 跳至導覽 跳至搜尋 世紀： 1世紀 2世紀 3世紀 年代： 160年代 170年代 180年代 190年代 200年代 210年代 220年代 年： 194年 195年 196年 197年 198年 199年 200年 201年 202年 203年 204年 干支年： 己卯年 （ 兔年 ） 年號： 199年 係平年。 目錄 1 大件事 2 到箇年出世嗰人 3 到箇年死嗰人 4 節慶、風俗習慣 大件事 199...
                                Read more