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Questions tagged [exponential-function]

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

7
votes
1answer
101 views

Find $x$ such that $(x^2 + 4x + 3)^x + (2x + 4)^x = (x^2 + 4x + 5)^x$

Find all $x \in (-1, +\infty)$ such that $(x^2 + 4x + 3)^x + (2x + 4)^x = (x^2 + 4x + 5)^x$. What I have done so far was a substitution $y = x + 2$ which results in a nicer form of the equation: $(...
7
votes
3answers
328 views

What are other solutions to this differential equation, “similar” to $\sin x$ and $e^x$?

I've been studying electronics, where they make great use of the relationship between the sine and exponential functions ($e^{i \omega t} = \cos{\omega t} + i \sin \omega t)$. This relationship is ...
6
votes
2answers
525 views

For which $x$ is $e^x$ rational? Transcendental?

Apart from the trivial cases, $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?
6
votes
1answer
300 views

Is there any nice explanation of why the complex exponential function has no roots in the complex plane? [duplicate]

Here I am not looking for an explanation that uses basic properties that complex exponential function has, such as $e^{z+w}=e^ze^w$ or $e^0=1$ or any other, if this fact can be explained by using ...
6
votes
2answers
185 views

Negative solution to $x^2=2^x$

Just out of curiosity I was trying to solve the equation $x^2=2^x$, initially I thought there would be just the two solutions $x=2$ and $x=4$, but wolfram shows that the two equations intersect at not ...
5
votes
3answers
743 views

Definition of $\exp(x)$

I've been thought at school that the definition of $\exp(x)$ is the unique function satisfying $\exp'(x)=\exp(x)$ and $\exp(0)=1$. But I've never found this definition somewhere else. On Wikipedia ...
5
votes
2answers
352 views

Solving $x^2 - 1 = e^x$

Can someone help me solve the equation $x^2 - 1 = e^x$ ? I tried taking the natural logarithm of both sides but I don't know where to go from there.. I got: $\ln(x^2 -1) = x$ But I don't know how ...
5
votes
2answers
317 views

Show that the series $\sum(\exp(\frac{(-1)^n}{n})-1)$converges, but not absolutely.

Show that the series converges, but not absolutely. $\sum_{n=1}^{\infty}( $exp$(\frac{(-1)^n}{n})-1)$. My Try: Let $a_n=$exp$(\frac{(-1)^n}{n})-1$. I was going to use alternating series test because ...
4
votes
4answers
2k views

Definition of exponential function, single-valued or multi-valued?

If we define $$e^z=1+z+\frac{z^2}{2!}+\cdots$$ then it is single-valued. However, if we write $$e^z=e^{z\ln e}$$ then it is multi-valued. Besides, $a^z$ is multi-valued in general. It is kind of ...
4
votes
3answers
97 views

Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1.

Show that $3^{2n+1}-4^{n+1}+6^n$ is never prime for natural n except 1. I tried factoring this expression but couldn't get very far. It is simple to show for even n but odd n was more difficult, at ...
4
votes
4answers
232 views

Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$ (x,y) = (1.82546...,1.82546..) $$ I expect there to be a solution ...
3
votes
3answers
239 views

How to show $\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$

I need to show the following: $$\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$$ For all complex numbers z. I don't know how to start this. Should I use l'Hopitals rule somehow?
3
votes
1answer
961 views

Proving that the exponential function is continuous

We aren't allowed to use many tricks such as difference quotient / integral calculus... Prove that $\exp$ is continuous at $x_{0}=0$ .....................................................................
3
votes
2answers
143 views

How can one find the zeroes of $f(x)=ae^{bx}+cx+d$?

A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form $$f(x)=ae^{bx}+cx+d = 0$$ where $a, b, c,$ ...
3
votes
3answers
2k views

Exponential function and uniform convergence of polynomials.

How can I prove that no sequence of polynomials converges uniformly to the exponential function? Thanks in advance for any help.
3
votes
3answers
110 views

Convergence of $\sum _{k=1}^\infty (1-\frac{1}{k})^{k^2}$ [closed]

Found the alternative form: $\sum _{k=1}^\infty ((1-\frac{1}{k})^{k})^k$. Tried various criteria, no luck so far.
3
votes
2answers
521 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
2
votes
3answers
191 views

Finding limit of $\left(1 +\frac1x\right)^x$ without using L'Hôpital's rule. [closed]

Is there a way to find this limit without using L'Hôpital's rule. Just by using some basic limit properties. $$\lim_{x\to\infty}\left(1+\frac1x\right)^x=e$$
2
votes
4answers
871 views

Incoherence using Euler's formula

Using the relation $\ e^{ix} = \cos(x) + i\sin(x)$ and substituting for $\ x = \pi$, we have the well-known Euler identity, $ e^{i\pi} = -1$. Substitute also for $ x = -\pi $, we have $ e^{-i\pi} = -1$...
2
votes
2answers
650 views

Express $-1+i$ in exponential form.

Express $-1+i$ in exponential form. My attempt so far Let $z=-1+i$ $$r=|z|=\sqrt2$$ $$\theta=\tan^{-1}(-1)=-\frac{\pi}{4}$$ Now, this is where I go wrong (I don't know why it's wrong!): So in ...
2
votes
5answers
930 views

Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$

This is easy to prove for negative $x$ but what about positive $x$? Should I use MVT?
2
votes
1answer
204 views

Is this a valid visualization of Euler's identity as a more generic pattern?

I was reading this nice question about a demonstration of Euler's identity, and tried to visualize how would look the left part of the identity in the complex plane by using the following function: ...
2
votes
4answers
236 views

How to prove that $\lim_{n\to\infty}\big(1+1/n\big)^n$ is equal to e

How do I prove the following limit without using the derivative. $$\lim_{n\to\infty}\big(1+1/n\big)^n$$ I have tried using the Binomial theorem but I haven't got too far. I have proved that the ...
2
votes
3answers
89 views

Determining $\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n$ with only elementary math

I am trying to find this limit: $$\lim_{n\to\infty}\left(n^{\tfrac{1}{n}}-1\right)^n,$$ I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of ...
2
votes
3answers
1k views

Is $x^x$ an exponential function?

I know that functions of the form $c^x$ are called exponential when $c$ is a constant. How about the function $x^x$? It seems somewhere in between exponential and double exponential to me. Is there a ...
2
votes
3answers
248 views

How to solve 5x=0.01^x

I just want to know how to solve: $$\ 5x=0.01^x$$ I have tried to use logarithms. It would be a huge help if someone could help because no matter what I do $\ x$ always gets stuck in a logarithm. ...
1
vote
3answers
612 views

Show an exponential function has a valid density.

Given: Let $X$ be exponential with parameter $\lambda$, that is $$ f_X(x) = \begin{cases} \lambda e^{-\lambda x} & \text{if }x> 0, \\ 0 &\text{for }x\leq 0. \end{cases} $$ where $\...
1
vote
1answer
597 views

Proof of existence and uniqueness of the exponential function using ODEs

In our lecture notes for our complex analysis class, we were given the following theorem: Theorem: There exists a unique complex function $f$ such that $f(z)$ is a single valued function $...
1
vote
2answers
122 views

Question about the proof of Central Limit Theorem

My instructor proved the central limit theorem using the characteristic function. I think the proof is a standard one because I found basically the same proof in wikipedia. So for i.i.d. ${X_1, X_2,\...
1
vote
2answers
312 views

Show that $\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}$ [duplicate]

Based on the definition of $e: = \lim_{x\to\infty} \left(1+\frac1x \right)^x$, how can we show that $$\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}?$$ So far I've tried ...
1
vote
2answers
95 views

how does one solve $\lim_{n\to \infty} (\frac{n^{2}-2n+1}{n^{2}-4n+2})^{n}$

I cant use lhopital's rule here is my attempt: Limit of $(\frac{n^{2}-2n+1}{n^{2}-4n+2})^{n}$ =$e^{\lim_{n\to \infty} n\times \ln(\frac{1 -2-n +\frac{1}{n^{2}}}{1 -\frac{4}{n} +\frac{2}{n^{2}}})}$ I ...
1
vote
1answer
65 views

Solution of trigonometric equation $0 = x\cos(x)+2$

I'm having a hard time finding the solution of the following equation: $$0 = x\cos(x)+2$$ This is part of showing that $f(x):= x^2e^{\sin(x)}$ has 3 extremas in $[-\frac{\pi}{2},\frac{\pi}{2}]$.
0
votes
1answer
126 views

Series proof for $e^x$.

Problem: Prove $$\sum_{n=0}^\infty \frac{1}{n!}x^n=e^x$$ I am a bit confused on how I should start this proof. Any pointers on how I should start would help.
0
votes
0answers
71 views

Solving an exponential inequality problem

How do I prove the following inequality : $$\Bigg(\frac{2}{\alpha^2} \, \big( e^{\alpha x} - e^{\alpha y} \big) \, + \, e^{\alpha y} (y^2 - x^2) \; \Bigg) > 0 $$ given, $x, y > 0$ ? Can ...
0
votes
2answers
80 views

Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$.

Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$ where $B$ is a bounded subset of $\mathbb{C}$. I have no idea how to even attempt this question so I would ...
0
votes
2answers
136 views

The problem of x = ln(x)

I am trying to find x values for points along the normal distribution curve, and I ended up with a problem that goes back to the method of solving $x = \ln x$. Right now, I have $\ln(a \mu) - \ln(10) =...
0
votes
2answers
275 views

Solving $\sinh(ax) = bx$

I need an equation that expresses $x$ in function of $a$ and $b$. $$\sinh(ax) = bx$$ I'm a newbie in mathematics, and i don't know where to get help. I need to get this problem solved in order to ...
0
votes
2answers
137 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: What is the correct integral of $\frac{1}{x}$? After the algebra I said that 'This step of course gives the argument of $\log(x)$ the value $e$......
0
votes
0answers
121 views

Integral of the form $\int_{0}^{\infty}e^{-ax}e^{-x^{4}}dx$

I'm searching a closed form for the integral of the form : $$ \int_{0}^{\infty}e^{-ax}e^{-x^\frac{\alpha}{2}}dx $$ especially for $\alpha=8$ After several attempts, I find that the general form can ...
0
votes
1answer
2k views

Limits and exponential: $\lim\limits_{x \to\infty} (x-x^2 \ln(1+1/x))$ [duplicate]

Evalulate $\lim\limits_{x \to ∞} (x-x^2 \ln(1+1/x))$ I know the answer is 1/2 and we have to make use of the exponential function but i can't seem to simplify the expression to get the answer, please ...
0
votes
2answers
67 views

How create an exponential equation with points? (natural exponential)

How do I make exponential equation with points? Here are my points: x = 0, y = 0 x = 1, y = 4.75000 x = 2, y = 9.26250 x = 3, y = 13.54937 x = 4, y = 17.62191 ...
-3
votes
1answer
166 views

proving “e” irrational using convergent series [closed]

So my math professor in college gave us this problem: $$U_n=\sum_{k=1}^{k=n}\frac{1}{k!}$$ $$V_n=U_n+ \frac{1}{n\cdot n!}$$ The problem has two questions: 1- Prove that the two series are adjacent ...
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 ...
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 <...
57
votes
3answers
963 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 ...
20
votes
3answers
502 views

$\int_{- \infty}^{\infty} \frac{f(x)}{1+\exp{g(x)}}dx=\int_{0}^{\infty} f(x) dx$ for $f(x)=f(-x),~g(x)=-g(-x)$ - are there other formulas like that?

If $f(x)$ any even function, integrable on $(0,\infty)$ and $g(x)$ any odd function, then we have: $$\int_{- \infty}^{\infty} \frac{f(x)}{1+e^{g(x)}}dx=\int_{0}^{\infty} f(x) dx \tag{1}$$ The ...
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?
22
votes
1answer
613 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_{...
25
votes
7answers
614 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 $c$...