Prove thatthe following sequence is increasing e n = ( 1 + 1 n ) n , n ≥ 1 ;

Question

Prove thatthe following sequence is increasing      e          n        =            (      1      +              1        n            )              n        ,    n  ≥  1  ;

Joshua Nelson · Accepted Answer

In order to prove that the given sequence is strictly increasing, we are to demonstrate       e          n      +      1        &amp;gt;      e    n  :            (        1  +            1              n        +        1                                )                    n      +      1        &amp;gt;            (        1  +            1      n                          )              n    .Let&#039;s rewrite the inequality above as:            (                          1        +                              1                          n              +              1                                                  1        +                              1            n                                                  )              n    &amp;gt;            1              1        +                              1                          n              +              1                                            .The right-hand side equals            1              1        +                              1                          n              +              1                                            =                    n        +        1                    n        +        2              =  1  −            1              n        +        2              .Now, let&#039;s focus on the left-hand side:            (                          1        +                              1                          n              +              1                                                  1        +                              1            n                                                  )              n    =            (                          (        n        +        2        )                  /                (        n        +        1        )                    (        n        +        1        )                  /                n                                )              n    =            (                          n        (        n        +        2        )                    (        n        +        1                  )          2                                        )              n    =            (        1  −            1              (        n        +        1                  )          2                                        )              n    .By the Bernoulli&#039;s inequality, the following holds:            (        1  −            1              (        n        +        1                  )          2                                        )              n    ≥  1  −            n              (        n        +        1                  )          2                    Now it&#039;s purely technical to show the desired inequality  1  −            n              (        n        +        1                  )          2                      &amp;gt;  1  −            1              n        +        2              ,because            n              (        n        +        1                  )          2                      &amp;lt;            1              n        +        2              .

Prove that the following sequence is increasing e <mrow class="MJX-TeXAtom-ORD">

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