Find the limit of T n 5 n + 4

Question

Find the limit of             T      n              5      n      +      4

Braedon Rivas · Accepted Answer

Assuming we know that       U    n   converges (which is the premise of the question), we see that      lim          n      →      ∞            U          n      +      1        =      lim          n      →      ∞            U          n        .Then,      lim          n      →      ∞                          U        n            +      3              5      −              U        n              =      lim          n      →      ∞            U    n    .Then, in the limit,                    U        n            +      3              5      −              U        n              =      U    n    ,and solving for       U    n   gives       U    n    =  1 or       U    n    =  3. However, since       U    0    =  0  , it is clear that       U    n   cannot exceed 1 and hence,       lim          n      →      ∞            U    n    =  1Again, the premise of the question assumes that       lim          n      →      ∞                  T      n              5      n      +      4       converges. Then, we can write      lim          n      →      ∞                  T              n        +        1                    5      (      n      +      1      )      +      4        =      lim          n      →      ∞                  T      n              5      n      +      4        .Further, we see that, through the sum,      T          n      +      1        =      ∑          k      =      1              n      +      1            1                  U        k            −      3        =      (          ∑              k        =        1                    n                    1                        U          k                −        3              )    +      1                  U                  n          +          1                    −      3        =      T    n    +      1                  U                  n          +          1                    −      3        .We also know that       lim          n      →      ∞            U          n      +      1        =  1  , so       lim          n      →      ∞            1                  U                  n          +          1                    −      3        =  −      1    2    . Thus, in the limit, we have            T              n        +        1                    5      (      n      +      1      )      +      4        =                    T        n            −              1        2                    5      n      +      9        =            T      n              5      n      +      4        .Finally, solving for       T    n   gives      T    n    =  −      1    10    (  5  n  +  4  )  ,so      lim          n      →      ∞                  T      n              5      n      +      4        =  −      1    10    .

Find the limit of T n </msub> 5 n + 4 </mr

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