show that ( 1 + x n </mfrac> ) n </msup> &lt;

Briana Petty

Briana Petty

Answered question

2022-05-29

show that ( 1 + x n ) n < e x and e x < ( 1 x n ) n if x < n
If n is a positive integer and if x > 0,show that
( 1 + x n ) n < e x and that e x < ( 1 x n ) n if x < n
I proved the first one by the inequalities
( 1 + x n ) n = k = 0 n ( n k ) x k n k k = 0 n x k k ! < e x
the second one is equal to ( 1 x n ) n < e x but I feel I couldn't find a way to start.
In a previous exercise I proved
( 1 ) n e x < ( 1 ) n k = 0 n ( 1 ) k x k k !
it seems to have some relation with the second question.So I hope someone could give me a hint. Thanks in advance.

Answer & Explanation

vikafa4g

vikafa4g

Beginner2022-05-30Added 15 answers

By convexity (or by any of a number of other approaches, for example because log ( 1 + x ) x for every x > 1), the graph of the exponential is above its tangent at 0, that is,
For every real number   x ,
1 + x e x .
Apply this to x / n and raise to the power n, this yields the first inequality.
Apply this to x / n and raise to the power n, this yields ( 1 x / n ) n e x hence, composing by u 1 / u, the second inequality.

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