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

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38
votes
11answers
21k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
38
votes
4answers
2k 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
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 ...
32
votes
3answers
1k 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
849 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 ...
22
votes
7answers
3k views

Is $\exp(x)$ the same as $e^x$?

For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of ...
21
votes
13answers
2k 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 ...
21
votes
1answer
337 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) ...
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
382 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
325 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 ...
18
votes
3answers
333 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. ...
17
votes
10answers
2k 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 ...
16
votes
1answer
865 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, ...
15
votes
3answers
340 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
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
533 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!} \,+\, ...
14
votes
2answers
3k 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 ...
13
votes
4answers
720 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
444 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
6answers
1k views

Proving the inequality $e^{-2x}\leq 1-x$

How do I prove the inequality $e^{-2x}\leq1-x$ for $0\leq x\leq1/2$?
12
votes
8answers
7k 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.
12
votes
3answers
485 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
4answers
384 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 ...
12
votes
0answers
207 views

Integrate this monster

Can you please help me? I've been trying for some time now to integrate this: $$\int_0^\infty g^{-(a+1)} \; \exp\left\{-\left(\frac{b}{g} + \frac{1}{2} \sum_{i=1}^{n} ...
11
votes
4answers
2k views

Exponential growth of cow populations in Minecraft

Minecraft is a computer game where you can do many things including farm cows. When fed wheat, cows in Minecraft breed with each other in pairs and produce one baby per pair. After about 20 minutes ...
11
votes
6answers
368 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
1answer
120 views

Reason for LCM of all numbers from 1 .. n equals roughly $e^n$

I computed the LCM for all natural numbers from 1 up to a limit $n$ and plotted the result over $n$. Due to the fast-raising numbers, I plotted the logarithm of the result and was surprised to find a ...
11
votes
1answer
211 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 $n\times n$ ...
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
5answers
839 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
7answers
417 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
4answers
886 views

Confused about complex numbers

I am confused about something: \begin{eqnarray} (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray} e^{2 i \pi}=1~ and~ 1^{0.5}=1 \end{eqnarray} ...
10
votes
4answers
247 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: $$\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} \ \ \ \qquad \forall x>0.$$
10
votes
2answers
219 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
966 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
99 views

Why is the ratio of the number of terms needed to achieve successive integer values in the harmonic series approximately $e$?

Consider the harmonic series: $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5} + \cdots .$$ It takes $1$ term to achieve a partial sum of $1$, since $1$ is the first number. It takes $4$ terms to ...
10
votes
2answers
205 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 ...
9
votes
4answers
705 views

Proof of inequality $e^x + e^{-x} \leq 2e^{x^2}$

How would I prove that $$e^x + e^{-x} \leq 2e^{x^2}, \quad \text{for all real $x$}?$$ I narrowed it down to proving for $x \in (-1,1)$. I observed that for $(0,1)$ and for $(-1,0)$ I may need to ...
9
votes
2answers
191 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
176 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 ...
9
votes
7answers
574 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 ...
9
votes
1answer
105 views

Are these equations known?

Hello I found two equations that lead to constant e. I wonder if they are known. I think especially first one is most likely known but I couldn't find, it is hard to search google with all these ...
8
votes
8answers
372 views

Why is the differentiation of $e^x$ is $e^x$?

$$\frac{d}{dx} e^x=e^x$$ Please explain simply as I haven't studied the first principle of differentiation yet, but I know the basics of differentiation.
8
votes
5answers
230 views

Solving $2^{2x+1} - 2^{x+4} = 2^3 - 2^x$

$$2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$ How can I solve an exponential equation that has many terms as the one above. Include more than one method if available.
8
votes
3answers
412 views

Solution to differential equations $y(0)=1$ and $y^{(n)}=y+1$

When I was solving some differential equations, I asked myself the following: Is there a function has the following: $$y'=y+1$$ $$y''=y+1$$ $$y'''=y+1$$ $$......$$ $$......$$ If the initial value is ...
8
votes
3answers
490 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
6answers
176 views

Why is $\ln(x^x)=x\ln(x)$ valid?

I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.
8
votes
1answer
253 views

Evaluate $\int_{-1}^{1} \exp(x+e^{x})\,dx$

Evaluate $$\int_{-1}^{1} \exp({x+e^{x}})\,dx$$ where $\exp(x)=e^x$. Can anyone give me any tips on where to start with this? I've tried doing it be substitution, with $ u=e^x$ and ended up needing ...