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Questions tagged [exponential-function]

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

4
votes
3answers
784 views

Closed form for $n$-th derivative of exponential: $\exp\left(-\frac{\pi^2a^2}{x}\right)$

I need the closed-form for the $n$-th derivative ($n\geq0 $): $$\frac{\partial^n}{\partial x^n}\exp\left(-\frac{\pi^2a^2}{x}\right)$$ Thanks! By following the suggestion of Hermite polynomials: $$...
3
votes
2answers
349 views

show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$

I have derived the inequality if $k>1$, ${n(n-1)⋯(n-k+1)\over k!} ({1\over n})^k<{(n+1)n⋯(n-k+2)\over k!} ({1\over n+1})^k$ But, my problem is how to use this inequality to prove that if $n\...
2
votes
1answer
11k views

How would I create a exponential ramp function from 0,0 to 1,1 with a single value to explain curvature?

I need an exponential function that will take linear input from 0,0 to 1,1 and give me back an exponential shaped curve such that changes in X near the 0 point result in small increases in Y, but each ...
15
votes
3answers
517 views

$e^{\left(\pi^{(e^\pi)}\right)}\;$ or $\;\pi^{\left(e^{(\pi^e)}\right)}$. Which one is greater than the other?

$\newcommand{\bigxl}[1]{\mathopen{\displaystyle#1}} \newcommand{\bigxr}[1]{\mathclose{\displaystyle#1}} $ $$\large e^{\bigxl(\pi^{(e^\pi)}\bigxr)}\quad\text{or}\quad\pi^{\bigxl(e^{(\pi^e)}\bigxr)}$$ ...
7
votes
6answers
185 views

How to solve this limit: $\lim\limits_{x \to 0}\left(\frac{(1+2x)^\frac1x}{e^2 +x}\right)^\frac1x$

$$\lim\limits_{x \to 0}\left(\frac{(1+2x)^\frac{1}{x}}{e^2 +x}\right)^\frac{1}{x}=~?$$ Can not solve this limit, already tried with logarithm but this is where i run out of ideas. Thanks.
5
votes
2answers
420 views

Find the limit of: $\lim_{n\to\infty} \frac{1}{\sqrt[n+1]{(n+1)!} - \sqrt[n]{(n)!}}$

Could be the following limit computed without using Stirling's approximation formula? $$\lim_{n\to\infty} \frac{1}{\sqrt[n+1]{(n+1)!} - \sqrt[n]{(n)!}}$$ I know that the limit is $e$, but I'm ...
5
votes
4answers
1k views

Euler's formula, is this true? [duplicate]

*I've changed this question as below. Let me have a function such as $ f(k) = \exp(j 2 \pi k ) $, where $k$ is real value. Using Euler's formula, we can write $f(k)$ as below, $$ f(k) = \exp(j 2 \...
5
votes
2answers
193 views

The general solution of $x^a = a^x$ for real $a >0$

What are the roots of $$f(x) = x^a - a^x$$ for real $a > 0$? Case 1: For $0 < a < 1$ there is 1 solution, $x=a$. Case 2: For $1\le a < e$ there are 2 solutions: $x=a$ and $[x>a]$. Case ...
3
votes
1answer
708 views

If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute

Let $A,B$ be complex valued square matrices. If $\exp(t(A + B)) = \exp(tA) \exp(tB)$ for all $t \geq 0$ then $A,B$ commute. The converse of this statement can be an easy application of the Cauchy ...
11
votes
1answer
394 views

How to prove that: $19.999<e^\pi-\pi<20$?

I would like to know how to prove $$e^\pi-\pi\sim 20.$$ More precisely, I want to show by using only mathematical tools that, $$19.999<e^\pi-\pi<20$$ I have checked with online ...
9
votes
3answers
346 views

Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$

Prove the following inequality: $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$ This should be solved using Taylor series. I tried expanding the left to the 5th degree and the right site to the ...
7
votes
10answers
181 views

How to Find $ \lim\limits_{x\to 0} \left(\frac {\tan x }{x} \right)^{\frac{1}{x^2}}$.

Can someone help me with this limit? I'm working on it for hours and cant figure it out. $$ \lim_{x\to 0} \left(\frac {\tan x }{x} \right)^{\frac{1}{x^2}}$$ I started transforming to the form $ \...
6
votes
3answers
927 views

Prove that $e^{-A} = (e^{A})^{-1}$

Let $A, B \in R^{n \times n}$. Prove that $e^{-A} = (e^{A})^{-1}$. ($R$ is the real numbers) I've tried messing around with both sides, evaluated as sums. I just can't get the two to match up. Any ...
5
votes
4answers
193 views

A binomial inequality with factorial fractions: $\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+…+\frac{1}{n!}$

Prove that $$\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$$ for $n>1 , n \in \mathbb{N}$.
4
votes
4answers
325 views

Finding limit without using limit

If we have to find the value of $$ \lim_{x \to 0} \frac{e^x-1}{x}$$ I tried to solve this by using series i.e by expanding $e^x$ and got the result. But if there is another method to solve this
8
votes
3answers
5k views

Proof that $e=\sum\limits_{k=0}^{+\infty}\frac{1}{k!}$

How can it be proved that the Euler constant equals the limit of the sum of all $\frac{1}{k!}$ when $k$ goes from $0$ to $+\infty$ ?
5
votes
3answers
779 views

Why $e^{i(π/3)} \ne -1$? [closed]

I understand why $e^{i\pi} = -1$and as a result $e^{i2\pi}=\left(e^{i\pi}\right)^2=1.$ These results can be confirmed using Euler's formula But why does $e^{i\pi/3}\neq -1$ as we can write it $(e^{i\...
4
votes
3answers
385 views

Prove $ e^x = \exp(x) $ starting with their limits-based definitions

Let $$ e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^{n} $$ where $ n \in \mathbb{R} $. Let $$ \exp(x) = \lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^{n} $$ where where $ n \in \mathbb{R} $ ...
4
votes
1answer
2k views

Fourier transform of exponential of a function

I am wondering what $\mathcal{F}[\exp(f)]$ is in terms of $\mathcal{F}[f]$. The farthest I have got is using the series expansion of $\exp$, such that I end up with $\mathcal{F}[\exp(f)] = \sum_{k=0}^...
2
votes
6answers
149 views

How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$

How to prove $$1+x \leq e^x~\forall x \in \mathbb{R}$$ I'm stuck, I tried taking logs but didn't know how to proceed.
2
votes
5answers
193 views

What is a nice way to prove that : $\frac{t}{t+1} \le 1-e^{-t}\le \frac{2t}{1+t}$

I am trying to prove that for $t>0$$$\frac{t}{t+1} \le 1-e^{-t}\le \frac{2t}{1+t}$$ I know that Simplest or nicest proof that $1+x \le e^x$ taking $$e^{-x} \le \frac{1}{1+x} \implies 1-e^{-x}\ge \...
1
vote
2answers
78 views

What could the ratio of two sides of a triangle possibly have to do with exponential functions?

Name says it all. The two seem so unrelated? What's more, if I'm not mistaken the exponential version contains an imaginary part. I'm kind of ignorant about imaginary numbers, but does this mean that ...
4
votes
4answers
2k views

Prove $(1 + \frac{1}{n})^n$ is bounded above

I've checked similar questions on the site but couldn't find satisfactory solutions or hints. Also, is there a more general approach to proving whether a given sequence is bounded below or above?
3
votes
6answers
830 views

How to calculate $\lim_{x \to \infty} \left ( \frac{x+2}{x} \right )^{x}=e^{2}$

I know from an online calculator http://www.numberempire.com/derivatives.php that $\lim_{x \to \infty} \left ( \frac{x+2}{x} \right )^{x}=e^{2}$. How do you calculate this step by step?
2
votes
2answers
137 views

Rational Exponent

Is $a^{p/q}$ equal to $a^{2p/2q}$? Do we need to simplify $p/q$ to its lowest terms? I need a strict mathematical definition which proves one or another statement. For example: $(-8)^{1/3} = -2$ ...
2
votes
3answers
155 views

Graph of $(-1)^x$

What would the graph of $(-1)^{x}$ look like? I know that the value of the function alternates between $1$ and $-1$ when it is defined so I think it would just be points spread over the lines $x=1$ ...
2
votes
3answers
459 views

How many different definitions of $e$ are there?

It seems as though, in my analysis and calculus courses, in particular, a common cop-out when asked to prove an identity involving $e$, is the phrase "it's true by definition". So, I'm trying to find ...
1
vote
1answer
311 views

Accurately computing $e^x - e^{-x}$

If $x$ is in a small neighbourhood near zero, how can we accurately compute $e^x - e^{-x}$? We know that if $\Delta x$ is the absolute error on $x$, then the absolute error on $e^x$ is $e^x \Delta x$....
1
vote
1answer
708 views

Deriving a Quaternion Extension of Euler's Formula

According to Wikipedia: A rotation through an angle of $\theta$ around the axis defined by a unit vector $\vec{u} = (u_x, u_y, u_z) = u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}$ can be ...
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$...
16
votes
7answers
6k views

What are your favorite relations between e and pi? [closed]

This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for ...
15
votes
2answers
331 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\ \operatorname{Ei}^2(...
10
votes
3answers
16k views

Why does integrating a complex exponential give the delta function?

How come, when we integrate a complex exponential from $ -\infty $ to $ \infty $, we get a scaled delta function? $$ \begin{align} \int_{-\infty}^{\infty} e^{i k x} \; dk & = 2 \pi \delta \left ( ...
15
votes
2answers
437 views

Prove $\lim\limits_{n\to∞}{\sum\limits_{x=0}^n\binom nx(1+{\rm e}^{-(x+1)})^{n+1}\over\sum\limits_{x=0}^n\binom nx(1+{\rm e}^{-x})^{n + 1}}=\frac 13$

I am trying to find limit of the following function: $$ \lim_{n\rightarrow \infty}\frac{\sum\limits_{x = 0}^{n}\binom{n}{x}\left[1 + \mathrm{e}^{-(x+1)}\right]^{n + 1}}{\sum\limits_{x=0}^{n} \binom{n}...
13
votes
1answer
765 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 ...
21
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 ...
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 ...
13
votes
2answers
2k views

Proof that $e^x$ is a transcendental function of $x$?

Let a function $f(x)$ be algebraic if it satisfies an equation of the form $$c_n(x)(f(x))^n + c_{n-1}(x)(f(x))^{n-1} + \cdots + c_0(x)=0,$$ for $c_k(x)$ rational functions of $x$, and let $f$ be ...
9
votes
1answer
224 views

Prove that ${e\over {\pi}}\lt{\sqrt3\over{2}}$ without using a calculator.

I have been working on a known question for a long time (this is "Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator") during this time I realized the ${e\over {\pi}}\lt{\sqrt3\over{2}}$. ...
7
votes
4answers
241 views

Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$

Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$ It's easy to see that $x=0$ and $x=1$ are solutions but are these the only one? How do I demonstrate that? I've tried to write them ...
5
votes
4answers
2k views

Solve equation $\exp(ax)+\exp(bx)=1$

The equation is $$ \exp\left(ax\right)+\exp\left(bx\right)=1, $$ where $a$ and $b$ are known real constants, $x$ is unknown. I would like to have the solution in form of relatively known special ...
17
votes
9answers
4k views

Exponential Function as an Infinite Product

Is there any representation of the exponential function as an infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e. $$\mathrm e^x=\prod_{n=0}^\...
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, ...
10
votes
1answer
3k views

domain of $x^x$

What will be the domain of $f(x)=x^x$? I have asked this to some teachers, they say that the domain is set of all nonnegative real numbers. It is true that there are infinite negative numbers for ...
9
votes
3answers
366 views

Proof of the inequality $e^x\le e^{x^2} + x$ [duplicate]

The question is to prove the inequality $e^x\le e^{x^2} + x$. I tried the Taylor expansion like ${e^x} = 1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + ...$ and $x + {e^{{x^2}}} = 1 + x + \frac{...
7
votes
11answers
488 views

Why is $e$ the number that it is? [duplicate]

Why is $e$ the number that it is? Most of the irrational number that we learn about in school have something to do with geometry, like $\pi$ is the ratio of a circle's diameter to its circumference. ...
6
votes
2answers
299 views

Is $e^x=\exp(x)$ and why?

In the comments to this question a discussion came up wether we have $e^x=\exp(x)$ by definition and what the "correct" definition of $\exp(x)$ is. Building on that, I want to line out the problem ...
5
votes
2answers
179 views

Validity and Equivalence of two definitions of the real exponential function

The Problem : We state the following two definitions of the real exponential function from the Pr$\infty$fWiki page. We're interested in showing that the two definitions are valid $($i.e. the ...
3
votes
1answer
241 views

Limit of $\frac{x^y-y^x}{x^x-y^y} $ while $x\to y$

$$\lim_{x\to y}\frac{x^y-y^x}{x^x-y^y} $$ I've tried L'Hospital and this. Both doesn't tends to work. Please help someone.
7
votes
3answers
1k views

Limit involving $(\sin x) /x -\cos x $ and $(e^{2x}-1)/(2x)$, without l'Hôpital

Find: $$\lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$$ I have factorized it in this manner in an attempt to use the formulae. I have tried to use that ...