How Can I Prove (1+x/n^n < e^x)?

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Discussion Overview

The discussion centers around proving the inequality (1 + x/n)^n < e^x, exploring methods of proof, including logarithmic manipulation and series expansion. The scope includes mathematical reasoning and potentially involves concepts from calculus and analysis.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant suggests taking the logarithm of both sides as a starting point for the proof.
  • Another participant proposes expanding the left-hand side using the binomial theorem and comparing it to the power series for e^x, noting that the solution is more straightforward for x > 0.
  • The same participant mentions that for x < 0, the proof may be more complex and recommends working with terms two at a time.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the approach to take, and multiple methods are suggested without agreement on which is preferable.

Contextual Notes

There are unresolved steps regarding the application of logarithms and the specific handling of terms in the expansion for different values of x.

wowolala
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the quesion is below, show that
show that ( 1+ [tex]\frac{x}{n}[/tex] ) ^n < e^x

at the first, i take log on both sides,.. but i couldn't go further.

can someboday help me?

thx
 
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What do you get when you apply log to both sides?
 
wowolala said:
the quesion is below, show that
show that ( 1+ [tex]\frac{x}{n}[/tex] ) ^n < e^x

at the first, i take log on both sides,.. but i couldn't go further.

can someboday help me?

thx
Expand the lhs by the binomial theorem and take the power series for e^x. The solution becomes obvious for x>0. For x<0, it is a little trickier - work with terms 2 at a time.
 
Last edited:
thx so much
 

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