Proofs that e equals the series sum and that e^x differentiates to itself

[dextercioby]

I'm looking for proofs to the 2 following results.

Let

$$\displaystyle{e=: \lim_{n\rightarrow +\infty} \left(1+\frac{1}{n}\right)^n}$$

Show that:

1. Universality of e.

$$\sum_{k=0}^{\infty} \frac{1}{k!} = e$$

2. Derivative of $e^x$.

$$(e^x)' = e^x, ~ \forall x\in\mathbb{R}$$

Searching google didn't get me satisfactory results.

Could you, please, post or link to proofs ? Thank you!

 

[brydustin]

The classic definition is based on the fact that d(2^x)/dx ~ 0.693147 and d(3^x)/dx~1.098612 at x=0. Then you guess that there must be a number such that f'(0)=1 where f=a^x (it turns out a =e and is unique).

===
g(x)=b^x

g'(x) =lim h->0 b^(x+h)-b^x/h
=lim h->0 (a^x)(a^h -1)/h
=(a^x) lim h->0 (a^h -1)/h
and the derivative is where at x=0 we get
lim h->0 (a^h -1)/h = f'(0)

f ' (0) = (2^x)(a^h -1)/h ~ 0.693147
f ' (0) = (3^x)(a^h -1)/h ~1.098612
So it seems like there should be a number that converges to 1.
But I don't know the proof that you mentioned. Hope that helps somewhat.

 

[brydustin]

Also, if you assume the geometric definition of natural log and its derivative then the derivative of e^x can be computed from the fact that

d(ln(exp(x))/dx = (d/dx (e^x))*(1/e^x) = 1 >>> d/dx (e^x) = e^x
probably not the most satisfactory answer since it pushes off the proof to something contingent on the natural log. But its perfectly accurate.

Of course you can also try the taylor expansion but that's not really "first principals"

 

[D H]

$$\sum_{k=0}^{\infty} \frac{1}{k!} = e$$

Lemma:
$$\lim_{n\to\infty}\frac{n!}{n^k (n-k)!} = 1$$
Use the above in conjunction with the binomial expansion of
$$e=\lim_{n\to\infty}\left(1+\frac 1 n\right)^n$$

$$(e^x)' = e^x, ~ \forall x\in\mathbb{R}$$

You need a definition of exp(x) for this. One definition of exp(x) is that it is the function that is equal to its own derivative such that exp(0)=1. That makes the proof a bit too easy. Try using
$$\exp(x)=\lim_{n\to\infty}\left(1+\frac x n\right)^n$$

 

[deluks917]

First one follows from second if you use taylor series.