How to find the sum of $1+(1+r)s+(1+r+r^2)s^2+dots$?

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











up vote
7
down vote

favorite
4












I was asked to find the geometric sum of the following:



$$1+(1+r)s+(1+r+r^2)s^2+dots$$



My first way to solve the problem is to expand the brackets, and sort them out into two different geometric series, and evaluate the separated series altogether:



$$1+(s+rs+dots)+(s^2+rs^2+r^2s^2+dots)$$



The only problem is it doesn't seem to work, as the third term, $(1+r+r^2 +r^3)s^3$ doesn't seem to fit the separated sequence correctly.



Any help would be appreciated.




sequences-and-series geometric-series




share|cite|improve this question

















  • 2




    Hint: $1+r+r^2 + cdots + r^k = frac1-r^k+11-r$
    – achille hui
    18 hours ago














up vote
7
down vote

favorite
4












I was asked to find the geometric sum of the following:



$$1+(1+r)s+(1+r+r^2)s^2+dots$$



My first way to solve the problem is to expand the brackets, and sort them out into two different geometric series, and evaluate the separated series altogether:



$$1+(s+rs+dots)+(s^2+rs^2+r^2s^2+dots)$$



The only problem is it doesn't seem to work, as the third term, $(1+r+r^2 +r^3)s^3$ doesn't seem to fit the separated sequence correctly.



Any help would be appreciated.




sequences-and-series geometric-series




share|cite|improve this question

















  • 2




    Hint: $1+r+r^2 + cdots + r^k = frac1-r^k+11-r$
    – achille hui
    18 hours ago












up vote
7
down vote

favorite
4









up vote
7
down vote

favorite
4






4





I was asked to find the geometric sum of the following:



$$1+(1+r)s+(1+r+r^2)s^2+dots$$



My first way to solve the problem is to expand the brackets, and sort them out into two different geometric series, and evaluate the separated series altogether:



$$1+(s+rs+dots)+(s^2+rs^2+r^2s^2+dots)$$



The only problem is it doesn't seem to work, as the third term, $(1+r+r^2 +r^3)s^3$ doesn't seem to fit the separated sequence correctly.



Any help would be appreciated.




sequences-and-series geometric-series




share|cite|improve this question













I was asked to find the geometric sum of the following:



$$1+(1+r)s+(1+r+r^2)s^2+dots$$



My first way to solve the problem is to expand the brackets, and sort them out into two different geometric series, and evaluate the separated series altogether:



$$1+(s+rs+dots)+(s^2+rs^2+r^2s^2+dots)$$



The only problem is it doesn't seem to work, as the third term, $(1+r+r^2 +r^3)s^3$ doesn't seem to fit the separated sequence correctly.



Any help would be appreciated.





sequences-and-series geometric-series





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited 3 hours ago









Asaf Karagila

291k31400731




291k31400731









asked 18 hours ago









Loo Soo Yong

624




624







  • 2




    Hint: $1+r+r^2 + cdots + r^k = frac1-r^k+11-r$
    – achille hui
    18 hours ago












  • 2




    Hint: $1+r+r^2 + cdots + r^k = frac1-r^k+11-r$
    – achille hui
    18 hours ago







2




2




Hint: $1+r+r^2 + cdots + r^k = frac1-r^k+11-r$
– achille hui
18 hours ago




Hint: $1+r+r^2 + cdots + r^k = frac1-r^k+11-r$
– achille hui
18 hours ago










3 Answers
3






active

oldest

votes

















up vote
15
down vote













Your sum can be written as
$$sum_n=0^inftyleft(sum_k=0^nr^kright)s^n=sum_n=0^inftyfrac1-r^n+11-rs^n=frac11-rsum_n=0^inftys^n-
fracr1-rsum_n=0^infty(rs)^n.$$
Can you take it from here?



P.S. Here we are assuming that $|s|<1$, $|rs|<1$ and $rnot=1$. What happens when $r=1$?






share|cite|improve this answer



















  • 1




    There's a typo in the last term (there should be a minus sign instead of a plus).
    – yoann
    17 hours ago










  • @yoann Thanks for pointing out!!
    – Robert Z
    16 hours ago

















up vote
7
down vote













If you multiply your series by $r-1$, you get$$(r-1)+(r^2-1)s+(r^3-1)s^2+cdots,tag1$$which is the sum of$$r+r^2s+r^3s^2+cdots$$with$$-1-s-s^2-cdots$$The sum of the first series is $frac r1-rs$, whereas the sum of the second one is $-frac11-s$. Therefore,$$(1)=frac1r-1left(frac r1-rs-frac11-sright)=frac1(1-s)(1-rs).$$






share|cite|improve this answer



















  • 1




    @farruhota I've edited my answer. Thank you.
    – José Carlos Santos
    17 hours ago

















up vote
3
down vote













You expanded the brackets, but did not actually group:




$$1+(1+r)s+(1+r+r^2)s^2+...=1+(s+rscolorred+...)+(s^2+rs^2+r^2s^2colorred+...).$$




Here is the way to group:
$$1+(1+r)s+(1+r+r^2)s^2+...=1+(colorreds+colorgreenrs)+(colorreds^2+colorgreenrs^2+colorbluer^2s^2)+cdots=\
(1+colorreds+colorreds^2+cdots)+(colorgreenrs+colorgreenrs^2+cdots)+(colorbluer^2s^2+r^2s^3+cdots)=\
frac11-s+fracrs1-s+fracr^2s^21-s+cdots=\
frac11-s(1+rs+r^2s^2+cdots)=cdots$$
Can you finish?






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%2f2872738%2fhow-to-find-the-sum-of-11rs1rr2s2-dots%23new-answer', 'question_page');

    );

    Post as a guest

















    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    15
    down vote













    Your sum can be written as
    $$sum_n=0^inftyleft(sum_k=0^nr^kright)s^n=sum_n=0^inftyfrac1-r^n+11-rs^n=frac11-rsum_n=0^inftys^n-
    fracr1-rsum_n=0^infty(rs)^n.$$
    Can you take it from here?



    P.S. Here we are assuming that $|s|<1$, $|rs|<1$ and $rnot=1$. What happens when $r=1$?






    share|cite|improve this answer



















    • 1




      There's a typo in the last term (there should be a minus sign instead of a plus).
      – yoann
      17 hours ago










    • @yoann Thanks for pointing out!!
      – Robert Z
      16 hours ago














    up vote
    15
    down vote













    Your sum can be written as
    $$sum_n=0^inftyleft(sum_k=0^nr^kright)s^n=sum_n=0^inftyfrac1-r^n+11-rs^n=frac11-rsum_n=0^inftys^n-
    fracr1-rsum_n=0^infty(rs)^n.$$
    Can you take it from here?



    P.S. Here we are assuming that $|s|<1$, $|rs|<1$ and $rnot=1$. What happens when $r=1$?






    share|cite|improve this answer



















    • 1




      There's a typo in the last term (there should be a minus sign instead of a plus).
      – yoann
      17 hours ago










    • @yoann Thanks for pointing out!!
      – Robert Z
      16 hours ago












    up vote
    15
    down vote










    up vote
    15
    down vote









    Your sum can be written as
    $$sum_n=0^inftyleft(sum_k=0^nr^kright)s^n=sum_n=0^inftyfrac1-r^n+11-rs^n=frac11-rsum_n=0^inftys^n-
    fracr1-rsum_n=0^infty(rs)^n.$$
    Can you take it from here?



    P.S. Here we are assuming that $|s|<1$, $|rs|<1$ and $rnot=1$. What happens when $r=1$?






    share|cite|improve this answer















    Your sum can be written as
    $$sum_n=0^inftyleft(sum_k=0^nr^kright)s^n=sum_n=0^inftyfrac1-r^n+11-rs^n=frac11-rsum_n=0^inftys^n-
    fracr1-rsum_n=0^infty(rs)^n.$$
    Can you take it from here?



    P.S. Here we are assuming that $|s|<1$, $|rs|<1$ and $rnot=1$. What happens when $r=1$?







    share|cite|improve this answer















    share|cite|improve this answer



    share|cite|improve this answer








    edited 16 hours ago


























    answered 18 hours ago









    Robert Z

    83.2k953122




    83.2k953122







    • 1




      There's a typo in the last term (there should be a minus sign instead of a plus).
      – yoann
      17 hours ago










    • @yoann Thanks for pointing out!!
      – Robert Z
      16 hours ago












    • 1




      There's a typo in the last term (there should be a minus sign instead of a plus).
      – yoann
      17 hours ago










    • @yoann Thanks for pointing out!!
      – Robert Z
      16 hours ago







    1




    1




    There's a typo in the last term (there should be a minus sign instead of a plus).
    – yoann
    17 hours ago




    There's a typo in the last term (there should be a minus sign instead of a plus).
    – yoann
    17 hours ago












    @yoann Thanks for pointing out!!
    – Robert Z
    16 hours ago




    @yoann Thanks for pointing out!!
    – Robert Z
    16 hours ago










    up vote
    7
    down vote













    If you multiply your series by $r-1$, you get$$(r-1)+(r^2-1)s+(r^3-1)s^2+cdots,tag1$$which is the sum of$$r+r^2s+r^3s^2+cdots$$with$$-1-s-s^2-cdots$$The sum of the first series is $frac r1-rs$, whereas the sum of the second one is $-frac11-s$. Therefore,$$(1)=frac1r-1left(frac r1-rs-frac11-sright)=frac1(1-s)(1-rs).$$






    share|cite|improve this answer



















    • 1




      @farruhota I've edited my answer. Thank you.
      – José Carlos Santos
      17 hours ago














    up vote
    7
    down vote













    If you multiply your series by $r-1$, you get$$(r-1)+(r^2-1)s+(r^3-1)s^2+cdots,tag1$$which is the sum of$$r+r^2s+r^3s^2+cdots$$with$$-1-s-s^2-cdots$$The sum of the first series is $frac r1-rs$, whereas the sum of the second one is $-frac11-s$. Therefore,$$(1)=frac1r-1left(frac r1-rs-frac11-sright)=frac1(1-s)(1-rs).$$






    share|cite|improve this answer



















    • 1




      @farruhota I've edited my answer. Thank you.
      – José Carlos Santos
      17 hours ago












    up vote
    7
    down vote










    up vote
    7
    down vote









    If you multiply your series by $r-1$, you get$$(r-1)+(r^2-1)s+(r^3-1)s^2+cdots,tag1$$which is the sum of$$r+r^2s+r^3s^2+cdots$$with$$-1-s-s^2-cdots$$The sum of the first series is $frac r1-rs$, whereas the sum of the second one is $-frac11-s$. Therefore,$$(1)=frac1r-1left(frac r1-rs-frac11-sright)=frac1(1-s)(1-rs).$$






    share|cite|improve this answer















    If you multiply your series by $r-1$, you get$$(r-1)+(r^2-1)s+(r^3-1)s^2+cdots,tag1$$which is the sum of$$r+r^2s+r^3s^2+cdots$$with$$-1-s-s^2-cdots$$The sum of the first series is $frac r1-rs$, whereas the sum of the second one is $-frac11-s$. Therefore,$$(1)=frac1r-1left(frac r1-rs-frac11-sright)=frac1(1-s)(1-rs).$$







    share|cite|improve this answer















    share|cite|improve this answer



    share|cite|improve this answer








    edited 17 hours ago


























    answered 18 hours ago









    José Carlos Santos

    111k1695171




    111k1695171







    • 1




      @farruhota I've edited my answer. Thank you.
      – José Carlos Santos
      17 hours ago












    • 1




      @farruhota I've edited my answer. Thank you.
      – José Carlos Santos
      17 hours ago







    1




    1




    @farruhota I've edited my answer. Thank you.
    – José Carlos Santos
    17 hours ago




    @farruhota I've edited my answer. Thank you.
    – José Carlos Santos
    17 hours ago










    up vote
    3
    down vote













    You expanded the brackets, but did not actually group:




    $$1+(1+r)s+(1+r+r^2)s^2+...=1+(s+rscolorred+...)+(s^2+rs^2+r^2s^2colorred+...).$$




    Here is the way to group:
    $$1+(1+r)s+(1+r+r^2)s^2+...=1+(colorreds+colorgreenrs)+(colorreds^2+colorgreenrs^2+colorbluer^2s^2)+cdots=\
    (1+colorreds+colorreds^2+cdots)+(colorgreenrs+colorgreenrs^2+cdots)+(colorbluer^2s^2+r^2s^3+cdots)=\
    frac11-s+fracrs1-s+fracr^2s^21-s+cdots=\
    frac11-s(1+rs+r^2s^2+cdots)=cdots$$
    Can you finish?






    share|cite|improve this answer

























      up vote
      3
      down vote













      You expanded the brackets, but did not actually group:




      $$1+(1+r)s+(1+r+r^2)s^2+...=1+(s+rscolorred+...)+(s^2+rs^2+r^2s^2colorred+...).$$




      Here is the way to group:
      $$1+(1+r)s+(1+r+r^2)s^2+...=1+(colorreds+colorgreenrs)+(colorreds^2+colorgreenrs^2+colorbluer^2s^2)+cdots=\
      (1+colorreds+colorreds^2+cdots)+(colorgreenrs+colorgreenrs^2+cdots)+(colorbluer^2s^2+r^2s^3+cdots)=\
      frac11-s+fracrs1-s+fracr^2s^21-s+cdots=\
      frac11-s(1+rs+r^2s^2+cdots)=cdots$$
      Can you finish?






      share|cite|improve this answer























        up vote
        3
        down vote










        up vote
        3
        down vote









        You expanded the brackets, but did not actually group:




        $$1+(1+r)s+(1+r+r^2)s^2+...=1+(s+rscolorred+...)+(s^2+rs^2+r^2s^2colorred+...).$$




        Here is the way to group:
        $$1+(1+r)s+(1+r+r^2)s^2+...=1+(colorreds+colorgreenrs)+(colorreds^2+colorgreenrs^2+colorbluer^2s^2)+cdots=\
        (1+colorreds+colorreds^2+cdots)+(colorgreenrs+colorgreenrs^2+cdots)+(colorbluer^2s^2+r^2s^3+cdots)=\
        frac11-s+fracrs1-s+fracr^2s^21-s+cdots=\
        frac11-s(1+rs+r^2s^2+cdots)=cdots$$
        Can you finish?






        share|cite|improve this answer













        You expanded the brackets, but did not actually group:




        $$1+(1+r)s+(1+r+r^2)s^2+...=1+(s+rscolorred+...)+(s^2+rs^2+r^2s^2colorred+...).$$




        Here is the way to group:
        $$1+(1+r)s+(1+r+r^2)s^2+...=1+(colorreds+colorgreenrs)+(colorreds^2+colorgreenrs^2+colorbluer^2s^2)+cdots=\
        (1+colorreds+colorreds^2+cdots)+(colorgreenrs+colorgreenrs^2+cdots)+(colorbluer^2s^2+r^2s^3+cdots)=\
        frac11-s+fracrs1-s+fracr^2s^21-s+cdots=\
        frac11-s(1+rs+r^2s^2+cdots)=cdots$$
        Can you finish?







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered 17 hours ago









        farruhota

        13.4k2632




        13.4k2632






















             

            draft saved


            draft discarded


























             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2872738%2fhow-to-find-the-sum-of-11rs1rr2s2-dots%23new-answer', 'question_page');

            );

            Post as a guest
































































            Comments

            Post a Comment

            Popular posts from this blog

            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/… – Hector Blandin Nov 17 &
            Read more

            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