Show that the sequence $a_1=3$ and $a_{n+1}=frac{3(1+a_n)}{3+a_n}$ converges to $sqrt3$











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First of all let´s see how it goes: $3,frac{3(1+3)}{3+3}=frac{12}{6}=2,frac{3(1+2)}{3+2}=frac{9}{5}=1.8,...$ We see that it's decreasing.



What i need to show first is that the sequence decreases. To do that, i´d say:



$$a_n-a_{n+1}>0Rightarrow a_n-{frac{3(1+a_n)}{3+a_n}}>0Rightarrow a_n^2>3Rightarrow a_n>sqrt{3}$$ But i think this is like say that $a_n$ converges to $sqrt3$, which is asked to be proved. So, How can i show this sequence decreases?. And to show it's bounded?










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  • Still study the difference $a_n -sqrt 3$.
    – xbh
    Nov 16 at 2:15















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First of all let´s see how it goes: $3,frac{3(1+3)}{3+3}=frac{12}{6}=2,frac{3(1+2)}{3+2}=frac{9}{5}=1.8,...$ We see that it's decreasing.



What i need to show first is that the sequence decreases. To do that, i´d say:



$$a_n-a_{n+1}>0Rightarrow a_n-{frac{3(1+a_n)}{3+a_n}}>0Rightarrow a_n^2>3Rightarrow a_n>sqrt{3}$$ But i think this is like say that $a_n$ converges to $sqrt3$, which is asked to be proved. So, How can i show this sequence decreases?. And to show it's bounded?










share|cite|improve this question
























  • Still study the difference $a_n -sqrt 3$.
    – xbh
    Nov 16 at 2:15













up vote
1
down vote

favorite









up vote
1
down vote

favorite











First of all let´s see how it goes: $3,frac{3(1+3)}{3+3}=frac{12}{6}=2,frac{3(1+2)}{3+2}=frac{9}{5}=1.8,...$ We see that it's decreasing.



What i need to show first is that the sequence decreases. To do that, i´d say:



$$a_n-a_{n+1}>0Rightarrow a_n-{frac{3(1+a_n)}{3+a_n}}>0Rightarrow a_n^2>3Rightarrow a_n>sqrt{3}$$ But i think this is like say that $a_n$ converges to $sqrt3$, which is asked to be proved. So, How can i show this sequence decreases?. And to show it's bounded?










share|cite|improve this question















First of all let´s see how it goes: $3,frac{3(1+3)}{3+3}=frac{12}{6}=2,frac{3(1+2)}{3+2}=frac{9}{5}=1.8,...$ We see that it's decreasing.



What i need to show first is that the sequence decreases. To do that, i´d say:



$$a_n-a_{n+1}>0Rightarrow a_n-{frac{3(1+a_n)}{3+a_n}}>0Rightarrow a_n^2>3Rightarrow a_n>sqrt{3}$$ But i think this is like say that $a_n$ converges to $sqrt3$, which is asked to be proved. So, How can i show this sequence decreases?. And to show it's bounded?







sequences-and-series limits






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edited Nov 16 at 16:42

























asked Nov 16 at 2:10









Tom Arbuckle

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  • Still study the difference $a_n -sqrt 3$.
    – xbh
    Nov 16 at 2:15


















  • Still study the difference $a_n -sqrt 3$.
    – xbh
    Nov 16 at 2:15
















Still study the difference $a_n -sqrt 3$.
– xbh
Nov 16 at 2:15




Still study the difference $a_n -sqrt 3$.
– xbh
Nov 16 at 2:15










4 Answers
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Hint. If you can prove that it has some limit $L$ then you can take the limit of both sides of the recursive definition and conclude ...






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    Hint:




    • Let $f(x)=frac{3(1+x)}{3+x}$. Prove that $f(x) le x$ for $x ge sqrt 3$


    • Prove that $x ge sqrt 3 implies f(x) ge sqrt 3$



    Conclude that $a_n$ is decreasing and bounded below and hence converges. Since $f$ is continuous, $a_n$ converges to fixed point of $f$.






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      You can use induction to show $a_n>a_{n+1}$:
      $$1) a_1=3>2=a_2;\
      2) text{assume} a_{n-1}color{red}>a_n;\
      3) a_{n}=frac{3(a_{n-1}+1)}{3+a_{n-1}}=3-frac{6}{3+a_{n-1}}color{red}>3-frac6{3+a_n}=frac{3(a_n+1)}{3+a_n}=a_{n+1}.$$






      share|cite|improve this answer




























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        We have $a_1>0$ and from the definition of $a_{n+1}$ we have $a_n>0implies a_{n+1}>0 $ for any $n$. So by induction we have $a_n>0$ for all $nin Bbb N.$



        We have $a_1>sqrt 3$. If $a_n>sqrt 3$ then, since $a_{n+1}>0,$ we have $$a_{n+1}>sqrt 3iff a_{n+1}^2>3iff$$ $$iff frac {3^2(1+a_n)^2}{(3+a_n)^2}>3iff$$ $$iff 3^2(1+a_n)^2 -3(3+a_n)^2>0iff$$ $$iff 6(a_n^2-3)>0.$$ So by induction we have $a_n>sqrt 3$ for all $nin Bbb N.$



        Since $3+a_n>a_n>0$ we have $$a_n>a_{n+1} iff (3+a_n)left(a_n-frac {3(1+a_n)}{3+a_n}right)>0 iff$$ $$iff 3a_n+a_n^2> 3+3a_niff a_n^2>3,$$ and we have already shown that $a_n^2>3.$ So $a_n>a_{n+1}$ for all $n.$



        Since $(a_n)_n$ is a decreasing sequence of positive terms it has a limit $Lgeq 0$ . And we have $$L=lim_{nto infty}a_{n+1}=lim_{nto infty}frac {3(1+a_n)}{3+a_n}= frac {3(1+L)}{3+L}.$$ So $L=frac {3(1+L)}{3+L}, $
        which implies $L^2=3.$ Now $(Lgeq 0land L^2=3)implies L=sqrt 3.$






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          4 Answers
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          4 Answers
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          up vote
          1
          down vote













          Hint. If you can prove that it has some limit $L$ then you can take the limit of both sides of the recursive definition and conclude ...






          share|cite|improve this answer

























            up vote
            1
            down vote













            Hint. If you can prove that it has some limit $L$ then you can take the limit of both sides of the recursive definition and conclude ...






            share|cite|improve this answer























              up vote
              1
              down vote










              up vote
              1
              down vote









              Hint. If you can prove that it has some limit $L$ then you can take the limit of both sides of the recursive definition and conclude ...






              share|cite|improve this answer












              Hint. If you can prove that it has some limit $L$ then you can take the limit of both sides of the recursive definition and conclude ...







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Nov 16 at 2:15









              Ethan Bolker

              39.4k543102




              39.4k543102






















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                  Hint:




                  • Let $f(x)=frac{3(1+x)}{3+x}$. Prove that $f(x) le x$ for $x ge sqrt 3$


                  • Prove that $x ge sqrt 3 implies f(x) ge sqrt 3$



                  Conclude that $a_n$ is decreasing and bounded below and hence converges. Since $f$ is continuous, $a_n$ converges to fixed point of $f$.






                  share|cite|improve this answer



























                    up vote
                    0
                    down vote













                    Hint:




                    • Let $f(x)=frac{3(1+x)}{3+x}$. Prove that $f(x) le x$ for $x ge sqrt 3$


                    • Prove that $x ge sqrt 3 implies f(x) ge sqrt 3$



                    Conclude that $a_n$ is decreasing and bounded below and hence converges. Since $f$ is continuous, $a_n$ converges to fixed point of $f$.






                    share|cite|improve this answer

























                      up vote
                      0
                      down vote










                      up vote
                      0
                      down vote









                      Hint:




                      • Let $f(x)=frac{3(1+x)}{3+x}$. Prove that $f(x) le x$ for $x ge sqrt 3$


                      • Prove that $x ge sqrt 3 implies f(x) ge sqrt 3$



                      Conclude that $a_n$ is decreasing and bounded below and hence converges. Since $f$ is continuous, $a_n$ converges to fixed point of $f$.






                      share|cite|improve this answer














                      Hint:




                      • Let $f(x)=frac{3(1+x)}{3+x}$. Prove that $f(x) le x$ for $x ge sqrt 3$


                      • Prove that $x ge sqrt 3 implies f(x) ge sqrt 3$



                      Conclude that $a_n$ is decreasing and bounded below and hence converges. Since $f$ is continuous, $a_n$ converges to fixed point of $f$.







                      share|cite|improve this answer














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                      edited Nov 17 at 1:25

























                      answered Nov 16 at 18:01









                      lhf

                      161k9165384




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                          down vote













                          You can use induction to show $a_n>a_{n+1}$:
                          $$1) a_1=3>2=a_2;\
                          2) text{assume} a_{n-1}color{red}>a_n;\
                          3) a_{n}=frac{3(a_{n-1}+1)}{3+a_{n-1}}=3-frac{6}{3+a_{n-1}}color{red}>3-frac6{3+a_n}=frac{3(a_n+1)}{3+a_n}=a_{n+1}.$$






                          share|cite|improve this answer

























                            up vote
                            0
                            down vote













                            You can use induction to show $a_n>a_{n+1}$:
                            $$1) a_1=3>2=a_2;\
                            2) text{assume} a_{n-1}color{red}>a_n;\
                            3) a_{n}=frac{3(a_{n-1}+1)}{3+a_{n-1}}=3-frac{6}{3+a_{n-1}}color{red}>3-frac6{3+a_n}=frac{3(a_n+1)}{3+a_n}=a_{n+1}.$$






                            share|cite|improve this answer























                              up vote
                              0
                              down vote










                              up vote
                              0
                              down vote









                              You can use induction to show $a_n>a_{n+1}$:
                              $$1) a_1=3>2=a_2;\
                              2) text{assume} a_{n-1}color{red}>a_n;\
                              3) a_{n}=frac{3(a_{n-1}+1)}{3+a_{n-1}}=3-frac{6}{3+a_{n-1}}color{red}>3-frac6{3+a_n}=frac{3(a_n+1)}{3+a_n}=a_{n+1}.$$






                              share|cite|improve this answer












                              You can use induction to show $a_n>a_{n+1}$:
                              $$1) a_1=3>2=a_2;\
                              2) text{assume} a_{n-1}color{red}>a_n;\
                              3) a_{n}=frac{3(a_{n-1}+1)}{3+a_{n-1}}=3-frac{6}{3+a_{n-1}}color{red}>3-frac6{3+a_n}=frac{3(a_n+1)}{3+a_n}=a_{n+1}.$$







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                              answered Nov 17 at 3:33









                              farruhota

                              17.7k2736




                              17.7k2736






















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                                  0
                                  down vote













                                  We have $a_1>0$ and from the definition of $a_{n+1}$ we have $a_n>0implies a_{n+1}>0 $ for any $n$. So by induction we have $a_n>0$ for all $nin Bbb N.$



                                  We have $a_1>sqrt 3$. If $a_n>sqrt 3$ then, since $a_{n+1}>0,$ we have $$a_{n+1}>sqrt 3iff a_{n+1}^2>3iff$$ $$iff frac {3^2(1+a_n)^2}{(3+a_n)^2}>3iff$$ $$iff 3^2(1+a_n)^2 -3(3+a_n)^2>0iff$$ $$iff 6(a_n^2-3)>0.$$ So by induction we have $a_n>sqrt 3$ for all $nin Bbb N.$



                                  Since $3+a_n>a_n>0$ we have $$a_n>a_{n+1} iff (3+a_n)left(a_n-frac {3(1+a_n)}{3+a_n}right)>0 iff$$ $$iff 3a_n+a_n^2> 3+3a_niff a_n^2>3,$$ and we have already shown that $a_n^2>3.$ So $a_n>a_{n+1}$ for all $n.$



                                  Since $(a_n)_n$ is a decreasing sequence of positive terms it has a limit $Lgeq 0$ . And we have $$L=lim_{nto infty}a_{n+1}=lim_{nto infty}frac {3(1+a_n)}{3+a_n}= frac {3(1+L)}{3+L}.$$ So $L=frac {3(1+L)}{3+L}, $
                                  which implies $L^2=3.$ Now $(Lgeq 0land L^2=3)implies L=sqrt 3.$






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                                    We have $a_1>0$ and from the definition of $a_{n+1}$ we have $a_n>0implies a_{n+1}>0 $ for any $n$. So by induction we have $a_n>0$ for all $nin Bbb N.$



                                    We have $a_1>sqrt 3$. If $a_n>sqrt 3$ then, since $a_{n+1}>0,$ we have $$a_{n+1}>sqrt 3iff a_{n+1}^2>3iff$$ $$iff frac {3^2(1+a_n)^2}{(3+a_n)^2}>3iff$$ $$iff 3^2(1+a_n)^2 -3(3+a_n)^2>0iff$$ $$iff 6(a_n^2-3)>0.$$ So by induction we have $a_n>sqrt 3$ for all $nin Bbb N.$



                                    Since $3+a_n>a_n>0$ we have $$a_n>a_{n+1} iff (3+a_n)left(a_n-frac {3(1+a_n)}{3+a_n}right)>0 iff$$ $$iff 3a_n+a_n^2> 3+3a_niff a_n^2>3,$$ and we have already shown that $a_n^2>3.$ So $a_n>a_{n+1}$ for all $n.$



                                    Since $(a_n)_n$ is a decreasing sequence of positive terms it has a limit $Lgeq 0$ . And we have $$L=lim_{nto infty}a_{n+1}=lim_{nto infty}frac {3(1+a_n)}{3+a_n}= frac {3(1+L)}{3+L}.$$ So $L=frac {3(1+L)}{3+L}, $
                                    which implies $L^2=3.$ Now $(Lgeq 0land L^2=3)implies L=sqrt 3.$






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                                      up vote
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                                      We have $a_1>0$ and from the definition of $a_{n+1}$ we have $a_n>0implies a_{n+1}>0 $ for any $n$. So by induction we have $a_n>0$ for all $nin Bbb N.$



                                      We have $a_1>sqrt 3$. If $a_n>sqrt 3$ then, since $a_{n+1}>0,$ we have $$a_{n+1}>sqrt 3iff a_{n+1}^2>3iff$$ $$iff frac {3^2(1+a_n)^2}{(3+a_n)^2}>3iff$$ $$iff 3^2(1+a_n)^2 -3(3+a_n)^2>0iff$$ $$iff 6(a_n^2-3)>0.$$ So by induction we have $a_n>sqrt 3$ for all $nin Bbb N.$



                                      Since $3+a_n>a_n>0$ we have $$a_n>a_{n+1} iff (3+a_n)left(a_n-frac {3(1+a_n)}{3+a_n}right)>0 iff$$ $$iff 3a_n+a_n^2> 3+3a_niff a_n^2>3,$$ and we have already shown that $a_n^2>3.$ So $a_n>a_{n+1}$ for all $n.$



                                      Since $(a_n)_n$ is a decreasing sequence of positive terms it has a limit $Lgeq 0$ . And we have $$L=lim_{nto infty}a_{n+1}=lim_{nto infty}frac {3(1+a_n)}{3+a_n}= frac {3(1+L)}{3+L}.$$ So $L=frac {3(1+L)}{3+L}, $
                                      which implies $L^2=3.$ Now $(Lgeq 0land L^2=3)implies L=sqrt 3.$






                                      share|cite|improve this answer












                                      We have $a_1>0$ and from the definition of $a_{n+1}$ we have $a_n>0implies a_{n+1}>0 $ for any $n$. So by induction we have $a_n>0$ for all $nin Bbb N.$



                                      We have $a_1>sqrt 3$. If $a_n>sqrt 3$ then, since $a_{n+1}>0,$ we have $$a_{n+1}>sqrt 3iff a_{n+1}^2>3iff$$ $$iff frac {3^2(1+a_n)^2}{(3+a_n)^2}>3iff$$ $$iff 3^2(1+a_n)^2 -3(3+a_n)^2>0iff$$ $$iff 6(a_n^2-3)>0.$$ So by induction we have $a_n>sqrt 3$ for all $nin Bbb N.$



                                      Since $3+a_n>a_n>0$ we have $$a_n>a_{n+1} iff (3+a_n)left(a_n-frac {3(1+a_n)}{3+a_n}right)>0 iff$$ $$iff 3a_n+a_n^2> 3+3a_niff a_n^2>3,$$ and we have already shown that $a_n^2>3.$ So $a_n>a_{n+1}$ for all $n.$



                                      Since $(a_n)_n$ is a decreasing sequence of positive terms it has a limit $Lgeq 0$ . And we have $$L=lim_{nto infty}a_{n+1}=lim_{nto infty}frac {3(1+a_n)}{3+a_n}= frac {3(1+L)}{3+L}.$$ So $L=frac {3(1+L)}{3+L}, $
                                      which implies $L^2=3.$ Now $(Lgeq 0land L^2=3)implies L=sqrt 3.$







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                                      answered Nov 17 at 11:17









                                      DanielWainfleet

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