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

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1answer
30 views

Simplifying Exponentials (Fourier)

I am having trouble simplifying the following expression: $$ \frac{1}{7}\left(1+e^{-jk\frac{2\pi}{7}}+e^{-jk\frac{4\pi}{7}}+e^{-jk\frac{6\pi}{7}}+e^{-jk\frac{8\pi}{7}}\right) $$ I need to get $$ ...
1
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1answer
52 views

Asymptotics and little-o notation

I always have issues dealing with asymptotic notation... I am trying to verify the following step: $$\left(1-\frac{t^2}{2n} + o(1/n)\right)^n \to e^{-t^2/2}.$$ To change this into ...
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3answers
44 views

How to Differentiate $x^7(7x+5)^6$

I am trying to differentiate $f(x) = x^7(7x+5)^6$. So far I have done the following steps: 1) Use the product rule, which is $(x^7(6(7x+5)^5))+((7x^6)(7x+5)^6)$ 2) Factor out $x^6$ and $(7x+5)^5$ ...
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3answers
37 views

Initial values of a exponential decay

How can I estimate the initials values ($A$, $B$, $C$) of a exponential decay? I got the function and a set of experimental points. $p(t) = Ae^{-1.5t} + Be^{-0.3t} + Ce^{0.05t}$ $p(0.5)=6,\ ...
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1answer
47 views

Exponential Proof

Let $c(x)=\dfrac{3^x+3^{-x}}{2}$ and $s(x)=\dfrac{3^x-3^{-x}}{2}$. Show that $(c(x))^2=\frac{1}{2}(c(2x)+1)$. How does one go about solving this? I have honesty tried substituting in ...
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1answer
98 views

Radioactive Decay formula is $A=A_0e^{-kt}$. How many years until 10 grams decay so that only 8 remain

I have been trying this question for hours and come to a dead end every time... Consider the radioactive decay formula $A=A_0e^{-kt}$ where $A$ is the amount of radium remaining at the time $t$. ...
1
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1answer
27 views

Find the power series representation for $ f(x) = \arctan (e^x) $ and its interval of convergence

friends. As stated on the title, my question is: find the power series representation for $ f(x) = \arctan (e^x) $ and its interval of convergence. This question got me a bit confused due to the ...
4
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0answers
37 views

Which matrices $B \in M_2(\mathbb{R})$ can be written as $B = e^A$ for some $A \in M_2(\mathbb{R})$? [closed]

As the question title suggests, which matrices $B \in M_2(\mathbb{R})$ can be written as $B = e^A$ for some $A \in M_2(\mathbb{R})$? Thanks!
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1answer
33 views

Homework solving exponential equation, logarithmic equation and exponential equation.

I need help with three homework questions.¨ First one: $$\sqrt{3x^2-2x-15}=x+1$$ I don't know how to get the right answer. The answer is supposed to be 4. I get: $$ 3x^2-2x-15=(x+1)^2$$ $$ ...
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3answers
55 views

Question referring to the exponential function defined as a limit of a sequence

Suppose that $(x_n)_{n \in \mathbb{N}}$ is a sequence with $\lim_{n \to \infty} x_n = x \in \mathbb{R}$. I want to show that $$ \lim_{n \to \infty} \left( 1 + \frac{x_n}{n} \right)^n = \text{e}^x. $$ ...
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2answers
38 views

principal value of improper integrals

How can I find the principal value of the following? $$PV\int_{-\infty}^\infty \frac{e^{ix}}{x(x^2+x+1)} dx$$ I'm able to evaluate integrals which have trig identities in them or just polynomials ...
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2answers
28 views

Is $\int z^n e^{az}dz $ a combination of exponentials and polynomials?

We have $$I(n)=\int z^n e^{az}dz=\int z^n \left (\frac{1}{a}e^{az}\right )'dz=\frac{1}{a}z^ne^{az}-\frac{1}{a}\int nz^{n-1}e^{az}dz \\ \Rightarrow I(n)=\frac{1}{a}z^ne^{az}-\frac{1}{a}nI(n-1) \ \ ...
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3answers
89 views

Show that $e^x=\lim_{n\to\infty }(1+\frac{x}{n})^n$ using the fact that $e^q=\lim_{n\to\infty}(1+\frac{q}{n})^n$ for $q\in\mathbb Q$.

I have to show that $$\lim_{n\to\infty }\left(1+\frac{x}{n}\right)^n=e^x$$ for all $x\in\mathbb R$ using the fact that $$e=\lim_{n\to\infty }\left(1+\frac{1}{n}\right)^n.$$ I already showed that for ...
3
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1answer
89 views

Show that $\sum_{k=0}^\infty \frac{1}{k!}=e$ using $e=\lim_{n\to\infty }(1+\frac{1}{n})^n$

I have to show that $$\sum_{k=0}^\infty \frac{1}{k!}=e$$ using $$e=\lim_{n\to\infty }(1+\frac{1}{n})^n.$$ My attempt ...
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1answer
31 views

isolating a exponent from the sum of two terms ($16^x - 10^x = y$)

$16^x - 10^x = y$ How can I isolate x in this case? Not much other information to give, This is just an equation that I came up with, whilst messing around.
4
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3answers
55 views

Multivariable Exponential Limit Priblem

I am trying to find the following limit $$\lim \limits_{(x, y) \to (0, 0)} \frac{(e^x-1)(e^y-1)}{x+y}$$ I have tried approaching along the lines $y = -x + mx^2$ but I'm getting a zero instead of a ...
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0answers
97 views

How to prove $e^{1/e}$ is irrational?

How do we prove $e^{\frac{1}{e}}$ is irrational ? Also how do we show it is transcendental ? The number $\eta = \exp(\exp(-1))$ occurs naturally in the context of tetration and power towers. Let ...
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0answers
27 views

Wait time in queue for 2 server system (exponential process)

A customer has to be server 1 before being served by server 2. Service times are exponential with rates $\mu_1$ and $\mu_2$ respectively. After being done at server 1, you wait at that station till ...
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0answers
28 views

Memoryless property implies $E(\max\{X_1,X_2,\ldots,X_n\})- E(\min\{X_1,X_2,\ldots,X_n\})=E(\max\{X_1,X_2,\ldots,X_{n-1}\})$

$X_1,X_2,\ldots,X_n \sim \mathrm{Exp}(\lambda), \quad \text{i.i.d.}$ How to show the following equation by using memoryless property: $$E(\max\{X_1,X_2,\ldots,X_n\}) - E(\min\{X_1 , X_2, \ldots, ...
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2answers
38 views

Solving (easy) equation [closed]

$$41-(6x-11)^{\frac{2}{5}}=38$$ Can someone give me a detailed step-by-step solution.
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2answers
25 views

Type of equation that has the property that $g(z) = 1 - g(-z)$

I am currently working my way through logistic regression, which gives $g(z) = \frac{1}{1+e^{-z}}$ and it also says that $g(-z) = 1 - g(z)$. I understand how to manipulate the equations for this to be ...
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2answers
41 views

Solution of integral involving exponential and absolute values [closed]

I want to solve this integral $\int_0 ^\infty e^{-iwx}e^{-α|x|} dx$ Any ideas on how to solve it?
4
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3answers
100 views

Find $\sum_{n=1}^\infty \frac{n^3}{n!}$.

I have to find the sum of the following series: $$\sum\limits_{n=1}^\infty \frac{n^3}{n!}$$ I know how to prove the convergence of this series, but how do i find the sum. I can't use the properties ...
1
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1answer
29 views

Computing Lyapunov Exponents

Consider $\mathcal{T}=S^{1}\times D^{2}$, where $S^{1}=[0,1]\mod 1$ and $D^{2}=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}\le 1\}$. Fix $\lambda\in(0,\frac{1}{2})$ and define the map ...
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1answer
47 views

Prove $\frac {1}{1\cdot2} - \frac {1}{2\cdot3} + \frac{1}{3\cdot4}-\cdots = \ln 2 - 1$

Please help me with this problem. I have figured out the $n^{th}$ term as $\frac {(-1)^{n+1}}{n(n+1)}$. How do I proceed ?
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1answer
39 views

Problem understanding proof that $e^a+e^b \geq e^{a+b}$

The answer is shown here by Rajada: Proof for which exponent is greater q1.) His solution seems nice but I can't understand how he gets to the second line from the first? (I understand the first ...
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6answers
45 views

Solving exponential equations like $2^{2x} - 3 \cdot 2^x - 10 = 0$

I have two equations that I'm not able to solve. I know the answers, but I can't get to them. $$(a) \qquad 2^{2x} - 3 \cdot 2^x - 10 = 0$$ (Answer: $x = \frac{\log 5}{\log 2}$.) On a) I ...
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1answer
46 views

How can $e^x$ be restated for small $x$?

Suppose I have the following equation: \begin{equation} 1=\frac{S_0}{\gamma}(1-e^{-\gamma T_n})+\lambda\int^{t_{n+1}}_{t_n}e^{-\gamma(t_{n+1}-\tau)}g(\tau)d\tau. \end{equation} If I make two ...
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3answers
71 views

How to find the limit of trig functions in exponents?

In my Calculus course, I am studying exponential functions and their involvement in limits. I do not understand why the answer to the following problem is $0$. $$ \lim_{ x \to \frac{\pi}{2}+} e^{\tan ...
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0answers
56 views

Writing the CDF of the exponential distribution in terms of another variable

I have a standard exponential CDF: $F_X(x) = $ \begin{cases} 1-e^{\lambda x}, & \text{x $\ge$ 0} \\ 0, & \text{otherwise} \end{cases} $ Y = ((X-2)^2-1)^2\\X = 2\pm(1\pm \sqrt{Y})^{1/2} $ ...
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0answers
21 views

Evaluating the limit of an exponential expression

I want to evaluate: $$\lim_{\beta \rightarrow \infty} \frac{2e^{-(E-\epsilon)\beta}+2e^{-(2E+\Delta-2\epsilon)\beta}}{1 + 2e^{-(E-\epsilon)\beta}+e^{-(2E+\Delta-2\epsilon)\beta}}$$ I know: $\Delta ...
2
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2answers
46 views

How to prove $f_n(z)=\sum_{j=0}^n\frac{z^j}{j!}$ doesn't converge uniformly to $e^z$ on $\Bbb{C}$?

We showed in class that the functions $f_n(z)=\sum_{j=0}^n\frac{z^j}{j!}$ converge uniformly to $e^z = \sum_{n=0}^\infty\frac{z^n}{n!}$ in any disc $D(0;R)$ such that $R>0$. But does $(f_n)$ ...
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0answers
20 views

Population Growth-fraction exponential

Annual plants produce seeds that can over-winter for several years before germinating, but the "parent" plants do not survive themselves. An invasive annual produces S seeds in a growing season. Of ...
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2answers
111 views

Why the serving time (excluding waiting time) at bank counter can be modeled as exponential distribution?

Hi I read something saying the serving time has exponential dist. I understand that, if the process is Poisson process, the ...
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1answer
22 views

Rearranging exponential functions

Using two different strategies, I've derived an equation for a particular function $f(\phi)$. That equation is $$ f(\phi)=\frac{1}{1-e^{-T\gamma}}(1-e^{-T\gamma\phi}). $$ However, the paper whose ...
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1answer
23 views

Evaluate $\exp{(a/T)} / T$ for $T \rightarrow 0$

How do I evaluate the expression: $\exp{(a/T)} / T$ for $T \rightarrow 0$ If I use L'Hôpital's rule I just get: $\frac{\frac{d}{dT}\exp{(a/T)}}{\frac{d}{dT}T} = \frac{-a\exp{a/T}}{T^2}$ I know I ...
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1answer
44 views

Equation which should be independent of $x$

An equation has the following form: $$Ae^{-i D x} + Be^{-i E x} = Ce^{-i F x}$$ where $A,B,C \in \mathbb{C}$ and $D,E,F \in \mathbb{R}$ are all constants and $x \in \mathbb{R}$, while $i$ is the ...
2
votes
2answers
60 views

Show that $S(n)\leq5\log_2(2n)+7$

I'm asked to show that for any positive integer n, we have: $S(n)\leq 5\log_2(2n)+7$ Where $S(n)$ is a total function (spans from positive integers to real integers), such that $S(1) = 7$, and ...
11
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5answers
307 views

Show $\lim\limits_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log

Taking as our definition of exponentiation repeated multiplication (extended to real exponents by continuity), can we show that the limit $$\lim_{h\to 0}\dfrac{a^h-1}{h}$$ exists, without ...
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3answers
41 views

Show $\frac{1}{n - 1} \geq e^\frac{1}{n} - 1, for~n \gt 1$

I am to show that $\frac{1}{n - 1} \geq e^\frac{1}{n} - 1, for~n \in \mathbb{N}^+, n \gt 1$. I tried substituting using $e^x = \sum_{i=0}^{\infty} \frac{x^i}{i!}$, which gives: $\frac{1}{n-1} \geq ...
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2answers
52 views

How to integrate exponential * fraction

$$\int_{-\infty}^{\infty}-\frac{1}{2\pi(\omega^2+16)}e^{-i \omega t} d\omega$$ I tried using partial fractions to separate $1/(\omega^2+16)$ into $-1/(8*(4-i\omega))$ and $1/(8*(-4-i\omega))$ but ...
1
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1answer
32 views

Exponential distribution and density

Let X be a positive random variable with density $e^{−x}\, 1_{(0,\infty)}(x)$ (the exponential distribution). What is the density of $1/(1 + X)$? What I am getting is $P(1/1+X < x) = 1-P(X>1/x ...
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0answers
54 views

Exponential function with a “kink”?

For a schematic of a real-world system ($x$ axis is time and $y$ axis is a sudden deterioration of the state of some physical system), I can almost model this system as $f(x) = -e^x$. However, this ...
2
votes
1answer
59 views

Express an exponential integral in hypergeometric form

I am new to hypergeometric function. I am trying to express this: $$\int_{0}^{\infty}e^{-ax^k+bx}dx$$ in a hypergeometric form. I have read some reference, but I don't get it how to cope with this ...
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1answer
74 views

Write a differential equation for compound interest?

I'm being asked the following question: If $A(t)$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation satisfied by $A(t)$. (The initial ...
0
votes
1answer
23 views

permutations of binary sequences

What is the proof that there are $2^n$ distinct binary codes of length n I know this progression also applies to the decimal ($10^n$) and hex ($16^n$) systems but how can this be shown?
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0answers
24 views

Functions like EWMA

I have a problem in which a threshold triggers an event. I want to estimate that threshold dynamically as I get more information progressively. Exponentially Weighted Mean Average is one such ...
0
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1answer
12 views

Solving $\frac{dV}{dt} = \frac{I}{C}$ using the Laplace transform

I have the following equation for the evolution of the membrane potential ($V$) of a neuron: $$ \frac{dV}{dt} = [-g_L(V-V_{rest}) + I_{syn}(t) + I_0] / C. $$ According to Equation 2.13 of this ...
7
votes
2answers
71 views

I'm trying to calculate $e^{At}$. Where do I go wrong?

Let $$A=\begin{bmatrix}0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -4 & 0\end{bmatrix}$$ I want to determine $e^{At}$. I tried it using ...
-1
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1answer
28 views

Exponential function leads to unsolvable logarithm.

Problem: $f(t) = 0.75 * 10^{kt}$ where k is a constant. We know that $f(2)=3$. Find $f(3)$. My approach: Rewrite $f(2)$ as: $0.75*10^{2k}=3$ Try to solve for k, eliminate 0.75 first: $10^{2k}=4$ ...