Limit of a sequence of fractionsI have that: S n = b ⋅ S n − 1 b + S n − 1 + awhere n ∈ N + and S 1 = a + bSo, for S 2 we will get: S 2 = b ⋅ S 2 − 1 b + S 2 − 1 + a = b ⋅ S 1 b + S 1 + a = b ⋅ ( a + b ) b + ( a + b ) + aAnd for S 3 : S 3 = b ⋅ S 3 − 1 b + S 3 − 1 + a = b ⋅ S 2 b + S 2 + a = b ⋅ ( b ⋅ ( a + b ) b + ( a + b ) + a ) b + ( b ⋅ ( a + b ) b + ( a + b ) + a ) + aAnd so on. Now my question is: what happens when n → ∞?

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Limit of a sequence of fractionsI have that:      S    n    =            b      ⋅              S                  n          −          1                            b      +              S                  n          −          1                      +  awhere   n  ∈            N        +   and       S    1    =  a  +  bSo, for       S    2   we will get:      S    2    =            b      ⋅              S                  2          −          1                            b      +              S                  2          −          1                      +  a  =            b      ⋅              S        1                    b      +              S        1              +  a  =            b      ⋅      (      a      +      b      )              b      +      (      a      +      b      )        +  aAnd for       S    3  :      S    3    =            b      ⋅              S                  3          −          1                            b      +              S                  3          −          1                      +  a  =            b      ⋅              S        2                    b      +              S        2              +  a  =            b      ⋅              (                              b            ⋅            (            a            +            b            )                                b            +            (            a            +            b            )                          +        a        )                    b      +              (                              b            ⋅            (            a            +            b            )                                b            +            (            a            +            b            )                          +        a        )              +  aAnd so on. Now my question is: what happens when   n  →  ∞?

Mira Spears · Accepted Answer

Assume that       S    n    =            p      n              q      n       is associated with   (      p    n    ,      q    n    )  ∈            R        2  . The recurrence                    (1)                              S                      n                          =                              (            a            +            b            )                          S                              n                −                1                                      +            a            b                                              S                              n                −                1                                      +            b                              can be written in the following form                    (2)                              (                                                                      p                  n                                                                                                      q                  n                                                              )                =                  (                                                    a                +                b                                            a                b                                                                    1                                            b                                              )                          (                                                                      p                                      n                    −                    1                                                                                                                        q                                      n                    −                    1                                                                                )                    hence the closed form of both the   {      p    n        }          n      ≥      0       and the   {      q    n        }          n      ≥      0       sequences just depends on the diagonalization of   M  =      (                            a          +          b                          a          b                                      1                          b                      )  , whose eigenvalues are given by       ζ          ±        =            (      a      +      2      b      )      ±                        a          2                +        4        a        b              2  .In particular                    (3)                              S          n                =                              A                          ζ              +              n                        +            B                          ζ              −              n                                            C                          ζ              +              n                        +            D                          ζ              −              n                                          for some constants   (  A  ,  B  ,  C  ,  D  ) depending on the initial values. By taking the limit as   n  →  +  ∞, we simply get             A      C      . On the other hand, by assuming       S    n    →  L as   n  →  ∞, such limit has to fulfill                    (4)                    L        =        a        +                              b            L                                b            +            L                              hence the only chances are   L  =      1    2        (    a    ±                  a        2            +      4      a      b        )

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