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I recently was teaching my friend about the number $e$. I introduced him the number by using the compound interest thing . Then I wrote down the general result -$$\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$$ The he told me that yes it works for $n=10,100,200,1000$. Beyond that his computer couldn't check . So he asked me for a formal mathematical proof of it . I thought of one but then that proof had natural logarithms - meaning they involved the number $e$ .I want to know the different ways through which this results can be proved , but without any use of $e$.

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marked as duplicate by Guy Fsone, mechanodroid, user99914, Claude Leibovici limits Nov 11 '17 at 6:06

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    $\begingroup$ How can you prove the result "without any use of $e$" if the answer is itself $e$? $\endgroup$ – 6005 Dec 17 '13 at 9:13
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    $\begingroup$ How do you define $e$? Afaik, it's normally defined as the limit you want to prove. $\endgroup$ – Daniel R Dec 17 '13 at 9:13
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    $\begingroup$ Does he have a VIC 20? Google can calculate the value (approximately of course) for $n=100000000$ (and much higher values, I'm sure). $\endgroup$ – David Mitra Dec 17 '13 at 9:17
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    $\begingroup$ You can introduce it by series to him, consequently he may calculate it easier. $\endgroup$ – Farshad Nahangi Dec 17 '13 at 9:24
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Here you go limit proof of e.

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  • $\begingroup$ Good paper ! Thanks. $\endgroup$ – Claude Leibovici Dec 17 '13 at 9:45

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