Intereting Posts

Generators for the radical of an ideal
In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$
Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$
Prove previsibility and $E \le E$
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How to prove these two random variables are independent?
Quadratic Extension of Finite field
Why does this limit exist $x^{x}$
Extreme points of unit ball of Banach spaces $\ell_1$, $c_0$, $\ell_\infty$
Looking for intuition behind coin-flipping pattern expectation
how to prove this extended prime number theorem?
Prove $\int_0^{\pi/2}{\frac{1+2\cos 2x\cdot\ln\tan x}{1+\tan^{2\sqrt{2}} x}}\tan^{1/\sqrt{2}} x~dx=0$
What is a physical interpretation of a skew symmetric bilinear form?
Can the limit of a product exist if neither of its factors exist?
Questions about derivative and differentiation

My expanded question:

Is showing

$\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists

as $z$ goes through real values

the same as $\lim_{n \to \infty} (1+\frac{1}{n})^n$ exists

as $n$ goes through integer values?

If not,

how much additional work

is needed to make the two

equivalent?

I am asking this because

I had posted a question which stated

a proof that the limit exists

for integer values.

I just a few minutes ago

answered a question

which involved the limit

as the argument

took on real values

by linking to my earlier answer.

The answer is here:

What is the most elementary proof that $\lim_{n \to \infty} (1+1/n)^n$ exists?

- Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{\left(\frac{\pi}{\alpha^5}\right)}^\frac{1}{2}$
- At what point does exponential growth dominate polynomial growth?
- Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator
- General formula for the higher order derivatives of composition with exponential function
- $e^{\left(\pi^{(e^\pi)}\right)}\;$ or $\;\pi^{\left(e^{(\pi^e)}\right)}$. Which one is greater than the other?
- Limit with a big exponentiation tower

It then occurred to me

that this is not true

until it is proved that

the limit through integer values

is the same as

the limit through real values.

The proof through integer values

showed that

$(1+1/n)^n$

is an increasing sequence

and that

$(1+1/n)^{n+1}$

is an decreasing sequence,

but says nothing about

what happens between the

integer values.

If it can be shown

that

$(1+1/z)^z$

is an increasing function

of $z$,

that would be enough,

but I do not know of

a proof that is as elementary

as the proof that

$\lim_{n \to \infty}(1+1/n)^n$

exists.

The usual proofs I have seen

involve the power series

for

$\ln(1-z)$.

This can be proved in

a number of ways,

including starting with

$\ln'(z) = 1/z$.

I realize that

I am meandering,

so I’ll leave this

at this point.

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- Probability and Laplace/Fourier transforms to solve limits/integrals from calculus.
- Prove that $\lim f(x) =0$ and $\lim (f(2x)-f(x))/x =0$ imply $\lim f(x)/x =0$
- Help with $\lim_{x\rightarrow +\infty} (x^2 - \sqrt{x^4 - x^2 + 1})$
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- Does the series $\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?
- Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$
- What is $\lim_{n \to \infty} n a_n$?
- Evaluation of limit at infinity: $\lim_{x\to\infty} x^2 \sin(\ln(\cos(\frac{\pi}{x})^{1/2}))$
- Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt{\frac{1}{3}+\cdots+\sqrt{\frac{1}{n}}}}$

Let’s denote $a_n=(1+\dfrac1n)^n$ for the members of the sequence.

Because $0<n\le x<n+1$, then

$$

1<(1+\frac1{n+1})<(1+\frac1x)\le(1+\frac1n).

$$

Consequently we get the upper bound

$$

(1+\frac1x)^x\le(1+\frac1x)^{n+1}<(1+\frac1n)^{n+1}=a_n\cdot(1+\frac1n),

$$

and similarly the lower bound

$$

(1+\frac1x)^x\ge(1+\frac1{n+1})^n=a_{n+1}\cdot(1+\frac1{n+1})^{-1}.

$$

So $(1+\dfrac1x)^x$ is sandwiched between two consecutive entries of the sequence $(a_n)$ multiplied by terms $\to 1$.

To answer: Yes, the two limit problems are equivalent.

By the definition of limits.

$c = \lim_{z \to \infty} (1+\frac{1}{z})^z$ exists if and only if

$\forall (s_n)_{n\in\mathbb{N}}, \, \, \left(\lim_{n \to \infty}(s_n) = +\infty \right)\implies \left(s = \lim_{n \to \infty} (1+\frac{1}{s_n})^{s_n} \,\mbox{ exists and }s = c\right)$.

Let $0 < x_1 < x_2$. Then $0 < \frac{x_1}{x_2} < 1$, so by the Bernoulli’s inequality

$$\left( 1 + \frac{1}{x_1} \right)^{x_1} = \left( 1 + \frac{1}{x_1} \right)^{\tfrac{x_1}{x_2} \cdot \ x_2} < \left( 1 + \frac{x_1}{x_2} \cdot \frac{1}{x_1} \right)^{x_2} = \left( 1 + \frac{1}{x_2} \right)^{x_2}.$$

Hence the function $f(x) = \left(1+\frac{1}{x}\right)^x$ is increasing on $(0, \infty)$.

The fundamental difference in dealing with limits $\lim_{n \to \infty}(1 + (1/n))^{n}$ and $\lim_{z \to \infty}(1 + (1/z))^{z}$ is that the conception of the first limit is simpler. The function $f(n) = (1 + (1/n))^{n}$ gives rational values for positive integers $n$ and can be calculated via simple arithmetic.

When you deal with $g(z) = (1 + (1/z))^{z}$ for positive real number $z$, then things are bit complex. The concept of an irrational exponent needs some development. If one has a sound theory of irrational exponents (some approaches are in my blog posts) then the proof that this limit exists is simple (based on taking logs).

In short the limit dealing with $n$ requires just theorems about monotone bounded sequences whereas limit concerning $z$ needs in addition the theory of arbitrary real exponents.

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