I want to determine the limit of this succession, for this I suppose that n 2 ( n 3 n ) n ≤ n 2 ( 1 + n 3 n ) n ≤ n 2 ( 2 n 3 n ) n

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I want to determine the limit of this succession, for this I suppose that      n    2    (      n          3      n            )    n    ≤      n    2    (            1      +      n              3      n            )    n    ≤      n    2    (            2      n              3      n            )    n

toriannucz · Accepted Answer

This first comment no longer applies, the post was editedYour second inequality is not correct. Take for example n=2. Then      n    2              (                        1          +          n                          3          n                    )        n    =  1  ,              (                        2          n                          3          n                    )        n    =      4    9    ,and clearly   1  ≥      4    9  This is still relevantWhat you could do, however, is notice that you can rewrite it as      n    2              (                        1          +          n                          3          n                    )        n    =      n    2              (              1                  3          n                    +              1        3            )        n    =            n      2              3      n                  (      1      +              1        n            )        n    .Now notice that      lim          n      →      ∞                  n      2              3      n        =  0(I will assume you already know this), and      lim          n      →      ∞                  (      1      +              1        n            )        n    =  e  .Thus if you take the limit, you can split it up into a product of limits, and what you end up with is      lim          n      →      ∞            n    2              (                        1          +          n                          3          n                    )        n    =  0  ⋅  e  =  0.

woowheedr · Accepted Answer

You have       n    2              (                        1          +          n                          3          n                    )        n    =            n      2              3      n                  (      1      +              1        n            )        n  It is well-known that       lim          n      →      ∞                  (      1      +              1        n            )        n   converges to e and it can be shown that       lim          n      →      ∞                  n      2              3      n        =  0 (for example, by using L&#039;hospital&#039;s Rule twice).Therefore,       lim          n      →      ∞            n    2              (                        1          +          n                          3          n                    )        n    =  0

I want to determine the limit of this succession, for this I suppose that n 2 </msup>

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New Questions in Calculus 1