To find the limit of ( 1 + 1 n 2 ) ( 1 + 2 n 2 ) . . . ( 1 + n n 2 )

Question

To find the limit of   (  1  +      1          n      2        )  (  1  +      2          n      2        )  .  .  .  (  1  +      n          n      2        )

bluayu0y · Accepted Answer

You have proved that   x  −            x      2        2    ≤  ln  ⁡  (  x  +  1  )  ≤  xNow      ∏          i      =      1        n    (  1  +      i          n      2        )  =      e                  ∑                  i          =          1                n            ln      ⁡      (      1      +              k                  n          2                    )      From (1) we get ,      ∑          i      =      1        n    (      i          n      2        −            i      2              2              n        4              )  ≤      ∑          i      =      1        n    ln  ⁡  (  1  +      i          n      2        )  ≤      ∑          i      =      1        n    (      i          n      2        )            n      (      n      +      1      )              2              n        2              −            n      (      n      +      1      )      (      2      n      +      1      )              12              n        4              ≤  S  ≤            n      (      n      +      1      )              2              n        2            where       S    =          ∑              i        =        1            n        ln    ⁡    (    1    +          i              n        2              )  Taking       lim          n      →      ∞       on each side and using squeeze theorem, we get      1    2    ≤      lim          n      →      ∞        S  ≤      1    2  Therefore      lim          n      →      ∞            ∏          i      =      1        n    (  1  +      i          n      2        )  =      e                  lim                  n          →          ∞                    S        =      e

To find the limit of ( 1 + 1 n 2 </msup> </mfrac> ) ( 1

Answered question

Answer & Explanation

New Questions in Calculus 1