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I think that for positive natural numbers t and n we have

$$ \left(1+\frac{n}{t}\right)^t\ \le\ e^n\,. $$

  1. Is this true? I have constructed a proof of it (which would probably take some lengthy typesetting here, and further I believe that the inequality could be very standard), but I wish to have an independent confirmation from the folks here.

  2. If the answer to the previous question is "yes", is there a proof in the published, reviewed literature (book, article, thesis, etc.) that I can reference? We really don't want to show that we've reinvented the wheel.

EDIT: Since the answers confirmed the inequality, and there are even duplicate questions, I'm really interested in part 2 from now on.

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marked as duplicate by Martin R, Peter Foreman, callculus, Adrian Keister, Lee David Chung Lin Apr 15 at 17:00

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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You Can prove by so many methods that $\forall s\ge0, \log(1+s)\le s \tag{1}$

Now use $(1)$ with $s={n\over t}$ and take exponential of both side


$$ \log\big(1+\frac{n}{t}\big)\le \frac{n}{t}\iff t\log\big(1+\frac{n}{t}\big)\le n\iff \log\big(1+\frac{n}{t}\big)^t\le n\iff\left(1+\frac{n}{t}\right)^t\ \le\ e^n$$

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