Use comparison test to show $sumfrac{3^n+7^n}{3^n+8^n}$ converges.











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I want to manipulate it such that $frac{3^n+7^n}{3^n+8^n}leq x^n$ for some $x<1$






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    up vote
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    down vote

    favorite












    I want to manipulate it such that $frac{3^n+7^n}{3^n+8^n}leq x^n$ for some $x<1$






    sequences-and-series







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I want to manipulate it such that $frac{3^n+7^n}{3^n+8^n}leq x^n$ for some $x<1$






      sequences-and-series







      share|cite|improve this question













      I want to manipulate it such that $frac{3^n+7^n}{3^n+8^n}leq x^n$ for some $x<1$






      sequences-and-series




      sequences-and-series






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




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      asked 14 hours ago









      m.bazza

      827




      827






















          3 Answers
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          up vote
          2
          down vote













          Notice



          $$ frac{ 3^n + 7^n }{3^n + 8^n} leqfrac{3^n+7^n}{8^n} = left( frac{3}{8} right)^n + left( frac{7}{8} right)^n $$



          Should I say more?






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            up vote
            0
            down vote













            We have that eventually



            $$frac{3^n+7^n}{3^n+8^n}leq frac{(7.5)^n}{8^n}=left(frac{7.5}{8}right)^n$$



            as an alternative by limit comparison test with $sum frac{7^n}{8^n}$ indeed



            $$frac{frac{3^n+7^n}{3^n+8^n}}{frac{7^n}{8^n}}=frac{8^n(3^n+7^n)}{7^n(3^n+8^n)} to 1$$






            share|cite|improve this answer




























              up vote
              0
              down vote













              It is often much easier and quicker to show $|a_n| leq Ccdot x^n$ with a positive constant $C$.



              In your case:
              $$color{blue}{frac{3^n+7^n}{3^n+8^n}} = left( frac{7}{8} right)^nfrac{left( frac{3}{7} right)^n + 1}{left( frac{3}{8} right)^n + 1} color{blue}{<} left( frac{7}{8} right)^n cdot frac{frac{1}{2} + 1}{1} =color{blue}{frac{3}{2} cdot left( frac{7}{8} right)^n}$$






              share|cite|improve this answer





















                Your Answer





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                3 Answers
                3






                active

                oldest

                votes








                3 Answers
                3






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes








                up vote
                2
                down vote













                Notice



                $$ frac{ 3^n + 7^n }{3^n + 8^n} leqfrac{3^n+7^n}{8^n} = left( frac{3}{8} right)^n + left( frac{7}{8} right)^n $$



                Should I say more?






                share|cite|improve this answer

























                  up vote
                  2
                  down vote













                  Notice



                  $$ frac{ 3^n + 7^n }{3^n + 8^n} leqfrac{3^n+7^n}{8^n} = left( frac{3}{8} right)^n + left( frac{7}{8} right)^n $$



                  Should I say more?






                  share|cite|improve this answer























                    up vote
                    2
                    down vote










                    up vote
                    2
                    down vote









                    Notice



                    $$ frac{ 3^n + 7^n }{3^n + 8^n} leqfrac{3^n+7^n}{8^n} = left( frac{3}{8} right)^n + left( frac{7}{8} right)^n $$



                    Should I say more?






                    share|cite|improve this answer












                    Notice



                    $$ frac{ 3^n + 7^n }{3^n + 8^n} leqfrac{3^n+7^n}{8^n} = left( frac{3}{8} right)^n + left( frac{7}{8} right)^n $$



                    Should I say more?







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 14 hours ago









                    Jimmy Sabater

                    1,769216




                    1,769216






















                        up vote
                        0
                        down vote













                        We have that eventually



                        $$frac{3^n+7^n}{3^n+8^n}leq frac{(7.5)^n}{8^n}=left(frac{7.5}{8}right)^n$$



                        as an alternative by limit comparison test with $sum frac{7^n}{8^n}$ indeed



                        $$frac{frac{3^n+7^n}{3^n+8^n}}{frac{7^n}{8^n}}=frac{8^n(3^n+7^n)}{7^n(3^n+8^n)} to 1$$






                        share|cite|improve this answer

























                          up vote
                          0
                          down vote













                          We have that eventually



                          $$frac{3^n+7^n}{3^n+8^n}leq frac{(7.5)^n}{8^n}=left(frac{7.5}{8}right)^n$$



                          as an alternative by limit comparison test with $sum frac{7^n}{8^n}$ indeed



                          $$frac{frac{3^n+7^n}{3^n+8^n}}{frac{7^n}{8^n}}=frac{8^n(3^n+7^n)}{7^n(3^n+8^n)} to 1$$






                          share|cite|improve this answer























                            up vote
                            0
                            down vote










                            up vote
                            0
                            down vote









                            We have that eventually



                            $$frac{3^n+7^n}{3^n+8^n}leq frac{(7.5)^n}{8^n}=left(frac{7.5}{8}right)^n$$



                            as an alternative by limit comparison test with $sum frac{7^n}{8^n}$ indeed



                            $$frac{frac{3^n+7^n}{3^n+8^n}}{frac{7^n}{8^n}}=frac{8^n(3^n+7^n)}{7^n(3^n+8^n)} to 1$$






                            share|cite|improve this answer












                            We have that eventually



                            $$frac{3^n+7^n}{3^n+8^n}leq frac{(7.5)^n}{8^n}=left(frac{7.5}{8}right)^n$$



                            as an alternative by limit comparison test with $sum frac{7^n}{8^n}$ indeed



                            $$frac{frac{3^n+7^n}{3^n+8^n}}{frac{7^n}{8^n}}=frac{8^n(3^n+7^n)}{7^n(3^n+8^n)} to 1$$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 14 hours ago









                            gimusi

                            85.4k74293




                            85.4k74293






















                                up vote
                                0
                                down vote













                                It is often much easier and quicker to show $|a_n| leq Ccdot x^n$ with a positive constant $C$.



                                In your case:
                                $$color{blue}{frac{3^n+7^n}{3^n+8^n}} = left( frac{7}{8} right)^nfrac{left( frac{3}{7} right)^n + 1}{left( frac{3}{8} right)^n + 1} color{blue}{<} left( frac{7}{8} right)^n cdot frac{frac{1}{2} + 1}{1} =color{blue}{frac{3}{2} cdot left( frac{7}{8} right)^n}$$






                                share|cite|improve this answer

























                                  up vote
                                  0
                                  down vote













                                  It is often much easier and quicker to show $|a_n| leq Ccdot x^n$ with a positive constant $C$.



                                  In your case:
                                  $$color{blue}{frac{3^n+7^n}{3^n+8^n}} = left( frac{7}{8} right)^nfrac{left( frac{3}{7} right)^n + 1}{left( frac{3}{8} right)^n + 1} color{blue}{<} left( frac{7}{8} right)^n cdot frac{frac{1}{2} + 1}{1} =color{blue}{frac{3}{2} cdot left( frac{7}{8} right)^n}$$






                                  share|cite|improve this answer























                                    up vote
                                    0
                                    down vote










                                    up vote
                                    0
                                    down vote









                                    It is often much easier and quicker to show $|a_n| leq Ccdot x^n$ with a positive constant $C$.



                                    In your case:
                                    $$color{blue}{frac{3^n+7^n}{3^n+8^n}} = left( frac{7}{8} right)^nfrac{left( frac{3}{7} right)^n + 1}{left( frac{3}{8} right)^n + 1} color{blue}{<} left( frac{7}{8} right)^n cdot frac{frac{1}{2} + 1}{1} =color{blue}{frac{3}{2} cdot left( frac{7}{8} right)^n}$$






                                    share|cite|improve this answer












                                    It is often much easier and quicker to show $|a_n| leq Ccdot x^n$ with a positive constant $C$.



                                    In your case:
                                    $$color{blue}{frac{3^n+7^n}{3^n+8^n}} = left( frac{7}{8} right)^nfrac{left( frac{3}{7} right)^n + 1}{left( frac{3}{8} right)^n + 1} color{blue}{<} left( frac{7}{8} right)^n cdot frac{frac{1}{2} + 1}{1} =color{blue}{frac{3}{2} cdot left( frac{7}{8} right)^n}$$







                                    share|cite|improve this answer












                                    share|cite|improve this answer



                                    share|cite|improve this answer










                                    answered 3 hours ago









                                    trancelocation

                                    7,9311519




                                    7,9311519






























                                         

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