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

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0
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1answer
47 views

Express recurrence in closed form

I am having trouble understanding the process of expressing the following recurrence in its' closed form. First of all, I do not really understand what "closed form" means. If someone could elaborate ...
1
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1answer
39 views

Finding nth derivative of an exponential function and its value at the origin.

I have a function defined as $f(x) = e^{-\frac{1}{x^2}}, $if $ x\ne0$; $0$ if $x =0$. where $f:[0,\infty) \to \mathbb{R}$ I am asked to prove the following: (a) that the nth derivative is of the ...
1
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1answer
17 views

Application of Borel-Cantelli for sequence of two parameters

Let $(A_{m,\ell})_{\ell \geq 0, m \geq 0}$ be a sequence of events in some probability space. How to show by using Borel Cantelli that, if $$\sum_{\ell \geq 0, m \geq 0} P(A_{m,\ell}) < \infty,$$ ...
3
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2answers
46 views

Relation on $\int_1^x\exp{t^2}dt$

Could you give me some leads to show the following relation : $$\forall x>0, \int_1^x\exp{t^2}dt = \frac{1}{2x}\exp{x^2} + \frac{1}{4x^3}\exp{x^2} - \frac{3}{4}\mathbb{e}+ \frac{3}{4} \int_1^x ...
0
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1answer
58 views

If $x^a + x^b = x^c + x^d$ how do $a ,b , c , d$ relationship are?

I used to solved these equation style and it's accidentally found an answer from matching $a, b, c,$ and $d$ relationship when $x^a + x^b = x^c + x^d $ (I assume that $ab = cd$) and found that's ...
2
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2answers
48 views

Find all real $a$ such that $6a^2+3=9^a$

Find all real $a$ such that $6a^2+3=9^a$ The problem seems to be very easy, but now i can't see an easy way to find if there are other roots than $1$. Tried using the derivative but that didn't help ...
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1answer
61 views

Power series $e^{-x^2}$

How would I create a power series of $f(x)=e^{-x^2}$ around $x_0=1$ without using a Taylor series? I need to know this for my upcoming exam so I would be really grateful to anyone who could show me ...
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1answer
36 views

exponential term evaluation doesn't make sense in this example

I am studying for my final and doing some practice questions, but I am confused by something: Here the solution says k at 0 we get N/2, but there is no way that answer is correct. If k is at 0 the ...
0
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1answer
82 views

Prove that the function is of exponential order and proving in mathematics

I'm currently learning about the Laplace transform and in my textbook in college and I have this definiton: Function $f$ is of exponential order if there exist constants $M>0$ and $a$ such that ...
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5answers
232 views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
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3answers
77 views

The limit of $e^{\frac{1}{x^2 + y^2}}$ as $(x, y) \to (0, 0)$

$$\lim_{(x, y) \to (0, 0)} e^{\frac{1}{x^2 + y^2}}$$ This really should be a simple limit question, I've done similar things many times before, but I'm very out of practice with limits and cannot for ...
0
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1answer
15 views

The time to fix a TV,is an exponential random variable with parameter $\lambda=\frac{1}{2}$.What is the probability that a fix take more than 2 hours?

I got the following question to solve: The time to fix a TV in hours, is an exponential random variable with parameter $\lambda = \frac{1}{2}$. What is the probability that a fix take more than 2 ...
2
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1answer
39 views

Rewriting $e^{-a|t|}$

here I have to prove the fourier transform of $e^{-a|t|}$ , the beginning of the proof is to rewrite $e^{-a|t|}$ as: $e^{-at} U(t) + e^{at} U(-t)$, I know how to continue the proof starting from this ...
0
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3answers
34 views

How did we come to this inequality $|\frac{e^x-1}{x}-1|\leq|\sum_{n=1}^{\infty}{\frac{|x|^n}{(n+1)!}}|$

How did we come to this inequality $|\frac{e^x-1}{x}-1|\leq|\sum_{n=1}^{\infty}{\frac{|x|^n}{(n+1)!}}|$. Using $e^x=\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$. I have this inequality in the proof of ...
6
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2answers
72 views

Prove or disprove that $p_n > e^{p_n - p_{n-1}}$ for large enough $n$.

Let $p_n$ denote the $n$-th prime. Prove or disprove that for large enough $n$ we have $$p_n > e^{p_n - p_{n-1}}.$$ The inequality trivially holds for all the twin primes larger than $7$. With ...
0
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1answer
33 views

How to obtain and graph a function that first grows exponentially and then decays exponentially?

I would like to know what the equation of a curve is if it grows exponentially (let's say it doubles each time). This would be: $f(x)=2^x$ But then I would like the line to exponentially decay (let ...
0
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1answer
22 views

Rate of convergence of the difference of two exponentials

I would like to find the convergence rate of the following function: $$f(x) = |e^{-ax}-e^{-bx}|,$$ with $a,b>0$ and $x\to+\infty$. By finding the convergence rate, I mean finding the largest ...
1
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1answer
78 views

Is the exponential map for $\text{Sp}(2n,{\mathbb R})$ surjective?

For $\mathfrak{g} := {\mathfrak s}{\mathfrak p}(2n,\mathbb{R})$ and $G = \text{Sp}(2n,{\mathbb R})$, is the exponential map \begin{equation} \text{exp} : \mathfrak{g} \to G \end{equation} surjective? ...
3
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1answer
64 views

Exponential series is cosh(x), how to show using summation?

I want to show that $$\cosh(x) = \sum_{n=0}^{\infty} ‎\frac{(x)^{2n}‎}{(2n)!}‎ $$ I know that $cosh(x) = \frac{exp(x)+exp(-x)}{2}$ but i cant seem to get there from the original series. I know that ...
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3answers
81 views

Exponentiation of a $2\times 2$ matrix

We know: $$\exp(At)=I+ \sum^{\infty}_{n=1}\frac{A^nt^n}{n!}$$ Here $$A= \begin{pmatrix} 0 & 1 \\ -w^2 & 0\end{pmatrix}$$ is a $2\times 2$ matrix, $I$ is identity matrix. How to show: ...
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1answer
35 views

Inequality between factorial and exponential

Trying to find a nice way to simplify the question: Which is bigger 2000! or 1000^2000? I don't know what kind of reasoning I can apply here.
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0answers
27 views

Functions which can be solutions of exponential algebraic equations.

Can the equation $p(f,e^f)=0$ have a solution in $L^{\infty}([0,1])$? Here $p(x,y)$ is a polynomial with complex coefficients, and $L^{\infty}([0,1])$ is the space of Lebesgue measurable ...
0
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1answer
27 views

Find the root of C [duplicate]

Can u help me to find a root for C (except c = 0) in below equation. $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ by expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
3
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0answers
87 views

Solving an exponential equation without the quadratic formula

High school math student here. In my homework I was asked to solve $16^x +4^{x+1} - 3= 0$ and I used substitution to get $x=\log_4{(-2+\sqrt7)}$. However, this was in the chapter on ...
0
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1answer
30 views

Calculating growth rate

Let's say I want to have saved $200 in one year. The first week I afford to save $1. I'm curious to find out how the calculation would look like to understand the following: By how much would I ...
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4answers
49 views

Extending $2^n > n $ from set of natural to set of real numbers

I was given a task to prove that $2^n>n$ for any $n \in N \cup \{0\}$. I am aware that this can be solved by induction and that the solution is pretty easy but instead of meddling with induction ...
0
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1answer
24 views

Solving a binomial when one of the terms is in the form $e^x$

Say I have the function $y=4e^{-2x}-3x$. I can use a graphing calculator to approximately determine the roots, but how do I find an exact answer?
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0answers
26 views

Total demand for two different prices, where market shares are determinened by logit model

The setting is simple, i.e. formula for demand of service/product is linear $$ d = \alpha - \beta p $$ where $ \alpha $ is maximum demand, $ \beta $ is some coefficient, and $ p $ is price. There ...
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3answers
45 views

Simplify $\frac {3^{(-3+x)}6^{(3-x)}}{3\cdot4^x}$ [closed]

$$\frac {3^{(-3+x)}6^{(3-x)}}{3\cdot4^x}$$ What is the simplest form?
6
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2answers
147 views

What is the function $f(x)=x^x$ called? How do you integrate it?

For real numbers $x > 0$, the function $f(x)=x^x$ seems pretty cool. Is there a name for this function? It's obviously been studied before. It grows faster than exponential functions and ...
2
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1answer
39 views

How can I solve an exponential equation of the following type?

I have an equation of the form $$ \frac{a^x}{d_1^x} + \frac{b^{x/2}}{d_2^x} = 1, $$ which I have already rewritten to $$ a^xd_2^x+d_1^xb^{x/2}-d_1^xd_2^x = 0. $$ However, I seem to be stuck here. ...
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0answers
37 views

Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$

Can you help me find a root for $c$ in the equation below? $$ce^{-c}-{10\over5}(1-e^{-c})^2=0$$ By expanding this I got, $$ce^{-c}-2 + 4 e^{-c}-2e^{-2c}=0$$ now grouping, ...
1
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1answer
24 views

Combined Distribution of Random variable

How to compute $P[T1 \le T2 \le t]$ for T1, T2 is independent random variable with exponential distribution in terms of cmf, pdf of T1 and T2? Similarly for $P[T1 \le T2 \le T3.. \le t]$ ? I tried ...
0
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0answers
29 views

Solving an exponential equation in like terms

This one may be fairly easy, yet, for the life of me, I can't remember how to do it. I would like to solve this equation to express x in terms of ...
0
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1answer
23 views

Limit of complex exponential

The following is the characteristic function of a random variable $X_n$:$$\phi_{n}(t)=\frac{1-e^{it}e^{\frac{it}{n}}}{(n+1)\left(1-e^{\frac{it}{n}}\right)}$$ for $t \in \mathbb R$. I am trying to ...
5
votes
2answers
89 views

Proving uniqueness of $e$ [duplicate]

Let's define $e$ as the number $a$ such that $\frac {d}{dx} a^x = a^x$. I'm trying to prove that this $a$ has to be $e$. I don't see any way of proceeding from here except by the limit definition ...
2
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4answers
128 views

Exponential of matrix

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc.) to calculate this exponential $e^{At}$ with $$A=\begin{pmatrix} ...
3
votes
1answer
91 views

How can I solve for x where $10^{10000} = x^x$

I hope this is not too elementary a question to post on here. If so, apologies. I'm stumped how I would solve for x where $10^{10000} = x^x$. Thanks!
2
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2answers
111 views

exponential function, lie group homomorphism

Let $f: \mathbb{R} \to \mathbb{C}^*$ be a continuous map satisfying for all $x, y \in \mathbb{R}$: $f(x + y) = f(x)f(y)$. $f(x) = 1$ for all $t = 2\pi n, n \in \mathbb{Z}$. Show that there exists ...
0
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2answers
42 views

Simple Explanation needed

I have been struggling with basics lately, so for this problem is-The value of $e^{-\infty}$ is 0 because $\frac{1}{e^{\infty}}=\frac{1}{\infty}=0$. Am I right ?
2
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1answer
57 views

Uniform convergence to exponential exercise

Yesterday I encountered the following exercise in a tutorial sheet from the University of Lyon : define a sequence of functions $(f_n)$ (with $f_n:[0,\infty) \to {\mathbb R}$) by ...
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2answers
85 views

How does $\int_0^\infty e^{-t^4}dt = \Gamma (\frac{5}{4}) ?$

My text book claims that $$\int_0^\infty e^{-t^4}dt = \Gamma \left(\frac{5}{4}\right).$$ I fail to see this. By the definition of the gamma function we have $$\Gamma (z) = \int_0^\infty ...
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2answers
35 views

How can I evaluate this exponential equation with natural logarithm $6161.859 = 22000\cdot(1.025^n-1)$?

I'm trying to evaluate an exponential equation with natural logarithm, but I'm certainly doing something wrong, can someone explain me how would you solve it using natural logarithm? $$6161.859 = ...
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2answers
55 views

Find 2 imaginary numbers that have a cosine of 4, using $\cos z =\frac{e^{iz}+e^{-iz}}{2}$

Use the definition $$ \cos z =\frac{e^{iz}+e^{-iz}}{2} $$ to find $2$ imaginary numbers having a cosine of $4$. I tried two approaches, both of which ended in failure: $$ 8=e^{iz}+e^{-iz}\\ ...
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0answers
109 views

Math equation problem - fitting wallpapers on a wall

I am building up a java program but don't have the right idea on how to resolve its math problem. The tasks I am doing are: I have to cover the wall with wallpaper. The wall is "a"(input) meters ...
2
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1answer
63 views

On Properties of Exponentially Prime Numbers

A usual prime number is a number greater than $1$ which is not in the form of multiplication of two numbers greater than $1$. We may consider the following natural generalizations: $p>1$ is $+$ - ...
1
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1answer
39 views

How can I write in Landau notation (or the like) that $2^x/x$ rises almost as fast as $2^x$?

Since $2^x \not\in O(2^x/x)$, we do not have $O(2^x/x)=O(2^x)$. But since $x$ rises linearly and $2^x$ exponentially, $2^x/x$ rises almost as fast as $2^x$. Can I somehow express this in Landau ...
0
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1answer
20 views

Exponential ( Problem with the mathematical description)

Well, I know the mathematical description of e as $\lim_{n\to\infty}(1-\frac{m}{n})^{n}=e^{-m}$ ,but today in my statistical mechanics class, while calculating the volume of thin shell of thickness ...
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3answers
52 views

evaluate $\lim(1+3x)^{(1/2x)}$.

Evaluate $\lim(1+3x)^{(1/2x)}$ as x approches 0 We know that : $\lim(1+x)^{(1/x)} =e$ "as x approaches 0 " I know that we can manipulate this equation to get $e$ to some power and I tried so many ...
0
votes
1answer
31 views

An $\Bbb{R}\to\Bbb{R}$ function with two plateaus of different heights and a valley

I am looking for a $\Bbb{R}\to\Bbb{R}$ function $f$ with two plateaus of different heights and a valley. The function has a minimum for $x=a$ and $f(a)=b$. The first (the one for smaller $x$) ...