Find the limit of b n = ∑ i = 1 n 1 a i a i + 1

Question

Find the limit of       b    n    =      ∑          i      =      1        n        1                  a        i                    a                  i          +          1

Gloletheods6g · Accepted Answer

Let  α  =            3      −              5              2    ,    β  =            3      +              5              2  be the zeros of the equation       x    2    −  3  x  +  1  =  0. Then by using   α  +  β  =  3 and   α  β  =  1, we find that      a          n      +      1        −  α      a    n    =  α  (      a          n      +      2        −  α      a          n      +      1        )  .From this, we get                    a        2            −      α              a        1                            a        i                    a                  i          +          1                      =                    α                  n          −          1                    (              a                  i          +          1                    −      α              a        i            )                      a        i                    a                  i          +          1                      =            α              i        −        1                    a      i        −            α      i              a              i        +        1              .So by telescoping,      b    n    =      1                  a        2            −      α              a        1                  (          1              a        1              −                  α        n                    a                  n          +          1                      )    .Furthermore it is not hard to check that   (      a    i    ) is increasing, hence       a    i    ≥  3 for all i Together this and |α|&amp;lt;1, we conclude      lim          n      →      ∞            b    n    =      1          (              a        2            −      α              a        1            )              a        1              =      1          2              5              −      1    6    .

Find the limit of b n </msub> = <munderover> ∑ <mrow clas

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