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Questions tagged [exponential-function]

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

111
votes
15answers
19k views

The math behind Warren Buffett's famous rule – never lose money

This is a question about a mathematical concept, but I think I will be able to ask the question better with a little bit of background first. Warren Buffett famously provided 2 rules to investing: ...
107
votes
9answers
15k views

Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$

I was wondering the following. And I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function ...
86
votes
13answers
91k views

Do factorials really grow faster than exponential functions? [closed]

Having trouble understanding this. Is there anyway to prove it?
79
votes
25answers
14k 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 ...
79
votes
11answers
10k views

Comparing $\pi^e$ and $e^\pi$ without calculating them

How can I calculate, without calculator or similar device, the values of $\pi^e$ and $e^\pi$ in order to compare them?
76
votes
2answers
24k views

Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
75
votes
6answers
6k views

What's so “natural” about the base of natural logarithms?

There are so many available bases. Why is the strange number $e$ preferred over all else? Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
64
votes
5answers
8k views

If $a+b=1$ so $a^{4b^2}+b^{4a^2}\leq1$

Let $a$ and $b$ be positive numbers such that $a+b=1$. Prove that: $$a^{4b^2}+b^{4a^2}\leq1$$ I think this inequality is very interesting because the equality "occurs" for $a=b=\frac{1}{2}$ and ...
57
votes
13answers
52k views

How to prove that exponential grows faster than polynomial?

In other words, how to prove: For all real constants $a$ and $b$ such that $a > 1$, $$\lim_{n\rightarrow\infty}\frac{n^b}{a^n} = 0$$ I know the definition of limit but I feel that it's not ...
57
votes
3answers
955 views

Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?

I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any ...
56
votes
2answers
1k views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ $$u_2=v_2=1$...
51
votes
4answers
3k 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 ...
49
votes
18answers
7k views

Intuitive Understanding of the constant “$e$”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "...
45
votes
21answers
5k views

Could you explain why $\frac{d}{dx} e^x = e^x$ “intuitively”?

As the title implies, It is seems that $e^x$ is the only function whoes derivative is the same as itself. thanks.
40
votes
3answers
1k views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

I am interested about some infinite product representations of $\pi$ and $e$ like this. Last week I found this formula on internet $$\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{...
40
votes
2answers
1k views

From $e^n$ to $e^x$

Solve for $f: \mathbb{R}\to\mathbb{R}\ \ \ $ s.t. $$f(n)=e^n \ \ \forall n\in\mathbb{N}$$ $$f^{(y)}(x)>0 \ \forall y\in\mathbb{N^*} \ \forall x\in\mathbb R$$ Could you please prove that ...
37
votes
8answers
3k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
37
votes
17answers
4k views

Intuitive explanation for why $\left(1-\frac1n\right)^n \to \frac1e$

I am aware that $e$, the base of natural logarithms, can be defined as: $$e = \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$$ Recently, I found out that $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)...
36
votes
4answers
2k views

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, $$e^{\pi}-{\pi}^e=e^{f(e)}-{e}^{f(\pi)}...
34
votes
14answers
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 edit: . .&...
34
votes
3answers
1k views

Prove that $(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$

Prove that $f(x)=(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$ for all $x>0.$ We have $f(x)=f(\frac{1}{x}), f'(x)=-\frac{1}{x^2}f'(\frac{1}{x}),$ so we only need to prove $f'(x)>0$ for $0 < x <...
33
votes
11answers
4k views

How do I interpret Euler's formula? [duplicate]

I don't understand the formula at all: $$e^{ix} = \cos(x) + i \sin(x)$$ I've tried reading all sorts of webpages and answers on the subject but it's just not clicking with me. I don't understand how ...
33
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 even ...
31
votes
7answers
4k views

Why is $e^\pi - \pi$ so close to $20$?

$e^\pi-\pi\approx 19.99909998$ Why is this so close to $20$?
31
votes
4answers
16k 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 ...
30
votes
3answers
3k views

Can hyperbolic functions be defined in terms of trignometric functions?

For example, can $\sinh x$ be written as a function of $\sin x$? Another question, are hyperbolic functions dependent of their trigonometric correspondence in any way?
30
votes
7answers
3k views

Why does $e$ have multiple definitions?

The number $e$ seems to have multiple definitions: $$\lim \limits_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$ The unique number $a$ such that $\int_1^a\frac{1}{x} \, dx = 1$ The unique number $...
30
votes
3answers
3k views

How to verify if a curve is exponential by eyeballing?

A plane curve is printed on a piece of paper with the directions of both axes specified. How can I (roughly) verify if the curve is of the form $y=a e^{bx}+c$ without fitting or doing any quantitative ...
28
votes
7answers
4k views

Proving $\mathrm e <3$

Well I am just asking myself if there's a more elegant way of proving $$2<\exp(1)=\mathrm e<3$$ than doing it by induction and using the fact of $\lim\limits_{n\rightarrow\infty}(1+\frac1n)^n=\...
28
votes
9answers
14k 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 $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
27
votes
8answers
15k 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 ...
27
votes
5answers
52k views

Proof of the derivative of ln(x)

I'm trying to prove that $\frac{\mathrm{d} }{\mathrm{d} x}\ln x = \frac{1}{x}$. Here's what I've got so far: $$ \begin{align} \frac{\mathrm{d}}{\mathrm{d} x}\ln x &= \lim_{h\to0} \frac{\ln(x + h) ...
27
votes
2answers
935 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 $z\in \mathbb{C}$:...
27
votes
1answer
644 views

Is this formula for $\frac{e^2-3}{e^2+1}$ known? How to prove it?

I found an interesting infinite sequence recently in the form of a 'two storey continued fraction' with natural number entries: $$\frac{e^2-3}{e^2+1}=\cfrac{2-\cfrac{3-\cfrac{4-\cdots}{4+\cdots}}{3+\...
26
votes
6answers
24k views

A question comparing $\pi^e$ to $e^\pi$ [duplicate]

I was doing an algebra problem set following a chapter on logarithms and exponentiation, and it presented this "bonus question": Without using your calculator, determine which is larger: $e^\pi$ or ...
26
votes
5answers
7k views

“What if” math joke: the derivative of $\ln(x)^e$

Randall Munroe, the creator of xkcd in his latest book What if writes (p. 175) that the mathematical analog of the phrase "knock me over with a feather" is seeing the expression $ \ln( x )^{e}$. And ...
26
votes
5answers
43k views

Difference between power law distribution and exponential decay

This is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks like ...
25
votes
6answers
3k views

Prove $|e^{i\theta} -1| \leq |\theta|$

Could you help me to prove $$ |e^{i\theta} -1| \leq |\theta| $$ I am studying the proof of differentiability of Fourier Series, and my book used this lemma. How does it work?
25
votes
7answers
610 views

Is $e$ a coincidence?

$e$ has many definitions and properties. The one I'm most used to is $$\lim_{n\to \infty}\left(1+\frac{1}{n}\right)^n $$ If someone asked me (and I didn't know about $e$): Is there a constant $c$...
24
votes
1answer
646 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) $a\notin\mathbb{Z}...
24
votes
1answer
721 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$. This is Mathematica code that finds the root: ...
23
votes
3answers
680 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. $$\...
23
votes
1answer
4k views

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can ...
22
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 ...
21
votes
9answers
2k 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 factoring: $...
21
votes
6answers
2k views

Which of the following powers is bigger: $2^{41}$ or $3^{24}$?

Calculate the maximum between the given numbers $$ \max(2^{41},3^{24})\text{.}$$ I got stuck when I tried to decompose the exponent as $41$ is a prime number.
21
votes
1answer
581 views

Continuous function with infinitely many zeros

Let $f:[0;1]\rightarrow\mathbb{R}$ a continuous function. Let $Z$ denote the set of zeros of $f$. If $Z$ is finite, it's easy to prove that $\lim\limits_{n\rightarrow+\infty}\left|\displaystyle\int_{...
20
votes
10answers
2k views

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

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 ...
20
votes
1answer
2k 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, ...
20
votes
3answers
1k views

Nth derivative of $x^x$

A few days ago, I was wondering about a series expansion for $$f(x) = \ x^x$$ so I tried taking a couple derivatives. From these I was able to guess at some formulas for the general coefficients of ...