For question involving exponential functions and questions on exponential growth or decay.

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38
votes
4answers
1k views

If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?

It is known that if two matrices $A,B \in M_n(\mathbb{C})$ commute, then $e^A$ and $e^B$ commute. Is the converse true? If $e^A$ and $e^B$ commute, do $A$ and $B$ commute? Edit: Addionally, what ...
33
votes
11answers
15k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
31
votes
18answers
3k views

Simplest or nicest proof that $1+x \le e^x$

The elementary but very useful inequality that $1+x \le e^x$ for all real $x$ has a number of different proofs, some of which can be found online. But is there a particularly slick, intuitive or ...
28
votes
3answers
897 views

Limit with a big exponentiation tower

Find the value of the following limit: $$\huge\lim_{x\to\infty}e^{e^{e^{\biggl(x\,+\,e^{-\left(a+x+e^{\Large x}+e^{\Large e^x}\right)}\biggr)}}}-e^{e^{e^{x}}}$$ (original image) I don't ...
25
votes
3answers
834 views

If there are entire $G_k$s such that $f=\exp\circ\exp\circ\cdots \circ\exp\circ G_k$ ($k$ times), must $f$ be constant?

I am a French guest and I hope that my English isn't too bad... So here is my issue: I consider an entire function $f$ which satisfies the following property for each complex number $z\in ...
21
votes
13answers
1k views

Intuitive proofs that $\lim\limits_{n\to\infty}\left(1+\frac xn\right)^n=e^x$

At this link someone asked how to prove rigorously that $$ \lim_{n\to\infty}\left(1+\frac xn\right)^n = e^x. $$ What good intuitive arguments exist for this statement? Later ...
21
votes
9answers
1k views

Finding the limit of $\left(\frac{n}{n+1}\right)^n$

Find the limit of: $$\lim_{n\to\infty}\left(\frac{n}{n+1}\right)^n$$ I'm pretty sure it goes to zero since $(n+1)^n > n^n$ but when I input large numbers it goes to $0.36$. Also, when ...
20
votes
18answers
2k views

How to understand why $x^0 = 1$, where $x$ is any real number?

Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$ I understand that the ...
19
votes
3answers
357 views

How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
18
votes
4answers
314 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
17
votes
10answers
1k views

Why isn't $\lim \limits_{x\to\infty}(1+\frac{1}{x})^{x}= 1$?

Given $\lim \limits_{x\to\infty}(1+\frac{1}{x})^{x}$, why can't you reduce it to $\lim \limits_{x\to\infty}(1+0)^{x}$, making the result "$1$"? Obviously, it's wrong, as the true value is $e$. Is it ...
17
votes
3answers
225 views

What combinatorial quantity the tetration of two natural numbers represents?

Tetration is a generalization of exponentiation in arithmetic and a part of a series of other generalized notions, Hyperoperators. Consider $m\uparrow n$ denotes the tetration of $m$ and $n$. i.e. ...
16
votes
1answer
774 views

Can you raise $\pi$ to a real power to make it rational?

We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this: Take $(\sqrt{2})^{\sqrt{2}}$ If it's rational, ...
14
votes
3answers
2k views

Is this question too easy or am I getting it wrong?

In my homework, I am asked to find the limit $$\lim\limits_{x\to0}{\frac{x}{e^x}}$$ But obviously, you could just substitute $x = 0$: $$\lim\limits_{x\to0}{\frac{x}{e^x}} = ...
14
votes
3answers
336 views

Find $x$ from $3^x\cdot x^3 = 1$

I saw a question on internet, tried to solve but I can't: \begin{equation} 3^x\cdot x^3 = 1 \end{equation} I get $\ln$ function and made some equalization and I reached that: \begin{equation} ...
14
votes
3answers
450 views

Is there a simple proof that $e^2$ is irrational using a positional numeral system?

My favorite proof that $e$ is irrational goes something like this. Observe that we can write any real number $r$ as $$ a \,+\, \frac{b_2}{2} \,+\, \frac{b_3}{3!} \,+\, \frac{b_4}{4!} \,+\, ...
13
votes
4answers
654 views

Prove that e is irrational

Prove that e is an irrational number. Recall that e $=\displaystyle\sum_{n=0}^\infty\frac{1}{n!},\,\,$ and assume $\mathrm{e}$ is rational. Then $$\sum\limits_{k=0}^\infty \frac{1}{k!} = ...
13
votes
6answers
425 views

$\pi$ from the unit circle, $\sqrt 2$ from the unit square but what about $e$? [duplicate]

If one wants to introduce $\pi$ to a not mathematically savvy person, the unit circle would be a good choice. The unit square would be the way to go for $\sqrt 2$. But what about $e$? I've reviewed ...
12
votes
3answers
449 views

Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?

In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one? I can of course ...
12
votes
2answers
1k views

Integral of matrix exponential

Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem \begin{align*} \dot{x}(t) = A x(t), \quad x(0) = x_0 \end{align*} is given by $x(t) = \mathrm{e}^{At} x_0$. I am ...
12
votes
4answers
344 views

Euler's identity: why is the $e$ in $e^{ix}$? What if it were some other constant like $2^{ix}$?

$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that ...
11
votes
6answers
6k views

Solve $e^x+x=1$

This seems to have stumped even my TA, so I'm asking it here. Given $e^x + x = 1$, solve for $x$. I already know that the answer is zero, but have no idea how to get there.
11
votes
6answers
363 views

Given a matrix $A$, find $A^n$

Given the matrix $$A = \left[{9\atop20}{-4\atop-9}\right]$$ how do I find $A^7$ or $A^{54}$ or $A^{2008}$ (etc.) ? I know I need the eigenvalues of A, but I'm not sure what to do afterwards. Is the ...
11
votes
0answers
160 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
10
votes
2answers
1k views

Show that $ e^{A+B}=e^A e^B$

If $A$ and $B$ are $n\times n$ matrices such that $AB = BA$ (that is, $A$ and $B$ commute), show that $$ e^{A+B}=e^A e^B$$ Note that $A$ and $B$ do NOT have to be diagonalizable.
10
votes
7answers
414 views

Find $\lim_{x\to-\infty}{x+e^{-x}}$

I have this exercise in my worksheet: $$\lim_{x\to-\infty}{x+e^{-x}}$$ I am always ending up with $-∞+∞$ or $\frac{∞}{∞}$. It says the answer is $+∞$, but how can I get that?
10
votes
5answers
757 views

Solve the equation $2^x=1-x$

Solve the equation: $$2^x=1-x$$ I know this is extremely easy and I know the solution using graphical approach. Basically, I can see the solution, but I can't work it out algebraically.
10
votes
2answers
213 views

Conjectural closed form for $\int_0^\infty\sqrt[3]z\ \operatorname{Ei}^2(-z)\,dz$

While trying to answer the question "A closed form for $\displaystyle\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$", I came up with a conjecture: $$\int_0^\infty\sqrt[3]z\ ...
10
votes
2answers
485 views

Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series

In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following: $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over ...
10
votes
2answers
203 views

Prove that $(e+x)^{e-x}>(e-x)^{e+x}$

I get stuck with proving that $$(e+x)^{e-x}>(e-x)^{e+x}$$ for $x \in (0, e)$. All I know, is that it is doable with Jensen inequality, and I started with defining $$f(x)=(e+x)^{e-x}$$ and further ...
10
votes
1answer
190 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
9
votes
4answers
240 views

prove that $\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} , \ x>0$,

Can you show very easy methods? I hope I'll see many methods. Thank you everyone. Prove that: $$\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} \ \ \ \qquad \forall x>0.$$
9
votes
2answers
172 views

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

This comes from the comments section of this question here, credits Lucian. The statement is $$\sum_{n=1}^\infty\left(e-\sum_{k=0}^n\frac1{k!}\right)=1$$ This looks really interesting, so I was ...
9
votes
3answers
156 views

Show that $\,\,n!<\mathrm{e}\left(\frac{n}{2}\right)^n$

I'd like to prove that $\,\,n!<\mathrm{e}\left(\frac{n}{2}\right)^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots ...
8
votes
3answers
476 views

Evaluate a limit (probably involving L'Hôpital rule)

Evaluate the limit: $$\mathop {\lim }\limits_{x \to \infty } x\left( {{{\left( {1 + {1 \over x}} \right)}^x} - e} \right)$$ My attempts didn't yield a result. I'd be glad for a guidance. Thanks!
8
votes
7answers
351 views

How to prove continuity of $e^x$.

I simply want a proof that $e^x$ is continuous. I have never really been able to find something satisfying these points: $e$ is defined to be the limit $\lim_{n\to\infty}\left(1+{1\over ...
8
votes
2answers
226 views

Is this a new expression for $e$?

Let $f(x) = x^x$. Then, let us define a function $p(x)$ such that: $$p(x) = \frac {f(x+1)}{f(x)} - \frac {f(x)}{f(x-1)}$$ As the value of $x$ increases, $p(x)$ approaches $e$, or (I think), ...
8
votes
3answers
259 views

Proving that a definition of e is unique

We can define $e$ as the number such that $\lim_{h \to 0} \frac{e^h-1}{h}=1$. However, of course we can only define $e$ this way if it is unique, i.e., there is no other value $c$ for which that is a ...
8
votes
1answer
129 views

How can I compute this limit? [duplicate]

I have to compute $$ \lim_{n\to\infty} \exp(-n)\left(1+n+\frac{n^2}{2}+\ldots+\frac{n^n}{n!} \right)$$ I think the value is 1, but i don't know how to proof this. Do I have to estimate the remainder ...
8
votes
1answer
205 views

Closed form of $\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$

Could you please help me to solve this integration problem? $$\int_0^2\frac{1}{2+\sqrt{3\,e^x+3\,e^{-x}-2}}dx$$ Its approximate numeric value is $0.419197813818367...$, but I could not find an exact ...
7
votes
7answers
1k views

How to solve $e^{ix} = i$

I am taking an on-line course and the following homework problem was posed: $$e^{ix} = i$$ I have no idea how to solve this problem. I have never dealt with solving equations that have imaginary ...
7
votes
5answers
140 views

If $\frac{d}{dx}e{^x} = e{^x}$, then why does $\frac{d}{dx}e^{-14}$ = 0?

If $\frac{d}{dx}e{^x} = e{^x}$, then why does $\frac{d}{dx}e^{-14}$ = 0? Why doesn't $\frac{d}{dx}e^{-14}$ = $e^{-14}$? I don't understand.
7
votes
2answers
96 views

Is $\int_x^{\infty}e^{-\frac{t^2}{2}} < \frac{1}{x}e^{-\frac{x^2}{2}}$?

While solving a problem in real analysis, I got stuck. I need to prove $$\int_x^{\infty}e^{-\frac{t^2}{2}}dt < \frac{1}{x}e^{-\frac{x^2}{2}} $$ Clearly I have to use some kind of inequality, but ...
7
votes
3answers
387 views

How was $e$ first calculated?

I understand how $\pi$ is calculated, but I am interested in references that explain when and how the natural exponent $e$ was developed. What mathematical principles are behind the value of $e$?
7
votes
2answers
222 views

“Fun” question: anyone know why $e$ (Euler's Number) was chosen for wave functions?

First, let me say that this is merely something I have always wondered about, and can never seem to find a good reference for. I simply want to know... the geek in me. Why was $e$ (Euler's Number) ...
7
votes
5answers
214 views

Analytical solution to $a^x+b^x=x$

Maybe stupid question, but I am wondering. Is there an analytical solution to equation $$a^x+b^x=x$$ for general $a$, $b$. How should I tackle this problem, if I want to find at least one $x$. ...
7
votes
7answers
1k views

About $\lim \left(1+\frac {x}{n}\right)^n$

I was wondering if it is possible to get a link to a rigorous proof that $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
7
votes
2answers
108 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
7
votes
4answers
197 views

Integral of $\int_{y_1}^{y_2} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x$

Does the following integral have a closed form solution? $$ \int_{y_1}^{y_2} \exp\left(\, -\alpha x\,\right)\, x \sqrt{1-x^2}{\rm d}x $$ $$ 0< y_1 < 1 $$ $$ 0< y_2 < 1 $$ Or is there an ...
7
votes
1answer
109 views

Need your help with the integral $\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$.

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{dx}{e^{\,e^{-x}} \cdot e^{\,e^{x}}}$$