Equivalence of Two Forms of the Exponential Function

Question

One encounters the following definitions for $e^x$ \begin{align} e^x &= \sum_{n = 0}^\infty \frac{x^n}{n!} \\ e^x &= \lim_{n \to \infty} \left( 1 + \frac{x}{n}\right)^n \end{align} One can show that these two definitions are equivalent. For example, it can be shown that they both are the unique solution to the differential equation $f' = f$ with $f(0) = 1$.

My question, which is admittedly a bit vague, is why are these two definitions equivalent? How are the sum and the limit expressions related? Is there a direct way to transform one into the other?

Edit

In the accepted answer for the linked question, the chain of equivalences: series expression $\iff$ unique solution to differential equation $\iff$ limit expression is shown. My question is if there's a way to bypass "unique solution to differential equation" in this chain of equivalences.

Consider binomial expansion: $\left(1+\dfrac xn\right)^n=1+x+\dfrac{n(n-1)}{n^2}\dfrac{x^2}2+\dfrac{n(n-1)(n-2)}{n^3}\dfrac{x^3}6+...$ — J. W. Tanner, Aug 21 '19 at 21:50
Thank you for the linked post. Maybe this will clarify my question: using the enumeration given in the accepted answer of the linked post, can one show $3 \iff 4$ directly? The linked post only shows $3 \iff 1 \iff 4$. — Charles Hudgins, Aug 21 '19 at 21:52
@J.W.Tanner Ah, that makes sense. I assume you finish the proof by passing the limit on both sides? If you post it as an answer I'll accept it. Also, based on my above comment, does it seem fair not to mark the question as a duplicate? — Charles Hudgins, Aug 21 '19 at 21:53
@CharlesHudgins: Yes, I understand that the linked answer does not show a direct connection between the series expression and the limit expression, and that is what your are requesting — J. W. Tanner, Aug 21 '19 at 22:09
This answer addresses this question in answer to a similar question. — robjohn, Aug 21 '19 at 22:25
You're right. I'll accept it if you mark the question as a duplicate. — Charles Hudgins, Aug 21 '19 at 22:32

score 3 · Accepted Answer · answered Aug 21 '19 at 21:58

3

Hint:

Using the binomial expansion,

$\lim\limits_{n\to\infty}\left(1+\dfrac xn\right)^n=\lim\limits_{n\to\infty}\left(1+x+\dfrac{n(n-1)}{n^2}\dfrac{x^2}2+\dfrac{n(n-1)(n-2)}{n^3}\dfrac{x^3}6+...\right).$

answered Aug 21 '19 at 21:58

J. W. Tanner

60,406

This is exactly what I was looking for. Thank you. – Charles Hudgins Aug 21 '19 at 22:02
You're welcome. Thanks for the acknowledgement! – J. W. Tanner Aug 21 '19 at 22:05
I must say I have some concerns that the proof that the limit on the right is in fact $e^x$ wont be so easy, but I'll see if I can make it work. – Charles Hudgins Aug 21 '19 at 22:06

Equivalence of Two Forms of the Exponential Function

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