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

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2
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2answers
52 views

$\lim_{n\to \infty} (1+n+\cos n) ^\frac{1}{2n+n \sin n}$

While in class, we were proving a limit problem using the Squeeze Theorem, but when I was reviewing my notes, I came up with a problem,, The first question was to prove that $$\lim_{n\to \infty}(1+n)...
2
votes
2answers
70 views

Does this definition of $e$ even make sense?

This sprung from a conversation here. In Stewart's Calculus textbook, he defined $e$ as the unique solution to $\lim\limits_{h\to 0}\frac{x^h-1}{h}=1$. Ahmed asked how do you define $x^h$ is not by $\...
0
votes
2answers
44 views

What is $\frac{(-2)^{x}}{2^{x-1}}$

The title says it all: $$\frac{(-2)^{x}}{2^{x-1}}$$ How is this computed? I'm reviewing the finer points of exponents so a thorough explanation would be most appreciated!
-1
votes
1answer
28 views

Can anyone walk me through calculating the differential equation…

I need help solving this equation, i've attempted using numerous methods. But I'm give choices with square roots as an exponent of e, and I haven't been able to match any of them. $$ \frac{1}{6x}\...
0
votes
1answer
33 views

When will the population of a sample double (using dif-eq)?

I have the initial equation $$\frac{dP}{dt}=kp$$ where P is the population, t is time, and k is some positive constant. The rest of what I'm given is that P(0) = A, what is the time for the population ...
0
votes
2answers
15 views

Solution to initial condition problem

$y=-ln(1-e^{(t+c)})$ I'm trying to find the solution to the initial condition $y(0)=-ln2$ Isolate c $0=ln(2)-ln(1-e^c)$ $0=ln({2\over1-e^c})$ $-e^c=2-1$ $e^c=-1$ $c=0$ I can't figure out ...
1
vote
1answer
34 views

Probability of exponential growth event

Under the assumption of exponential growth of a population of cells, the population size at time $t$, $N(t)$, is: $$N(t) = N_0\exp(rt)$$ where $r$ is the rate of division and $t$ is time. What is ...
1
vote
1answer
16 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
1
vote
0answers
14 views

Rearranging summation terms including a complex exponential expression

I'm reading a paper on signal processing and having a hard time wrapping my head around a step the author takes. The signal of interest is defined as $r_k = e^{j(2\pi\Delta f k T_s + \theta)} + v_k$ ...
0
votes
1answer
46 views

Stuck solving $\ln(e^y-1)-y=t+c$ for $y$

I'm trying to solve for $y$ $\ln(e^y-1)-y=t+c$ $e^y-1=e^{(t+c+y)}$ $e^y=e^{(t+c+y)}+1$ $y=t+c+y+1$ Where am I going wrong?
1
vote
0answers
46 views

Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that ...
0
votes
0answers
21 views

Proving Exponential Convergence

Consider the function $\dot{x} = f(x,t)$. I want to show that if there exists a function $V(x,t)$ and some positive constants $h,\delta,k_1,k_2,$ and $k_3$ such that for all $x \in B(0,h)$ and for all ...
0
votes
0answers
29 views

Closed form roots of sum of exponential functions

Do anyone know a way to solve an equation like the following (over the complex numbers)? $1+2^z+3^z=0$ I certainly cannot. I've tried by hand, and by mathematica, but I can't figure it out. Thanks in ...
0
votes
1answer
31 views

Integration of complex exponential function over $\mathbb C$

Find the limit $$\lim_{z \to \infty}\int_{\mathbb C}|w|e^{-|z-w|^2}dA(w) $$ where A is area measure such that dA=rdrd$\theta$ Please help me, I did four page computation by changing to polar ...
0
votes
3answers
54 views

Solve for $x$ for the following exponential equation $2^{2x+1} = 3^{2x+1}$. What am I doing wrong?

$2^{2x+1} = 3^{2x+1}$ $2^1=3$? Why can't I take $\log_2$ of both sides ?
2
votes
1answer
52 views

Exponential equation on the set of real numbers

Solve the following equation on the set of real numbers: $8^x+27^x+2·30^x+54^x+60^x=12^x+18^x+20^x+24^x+45^x+90^x$ $x=1; x=0; x=-1$ are trivial solutions, but I'm stuck with proving that there are ...
0
votes
0answers
15 views

Bounds for infinite series involving exponentials

Let $$ S(a,b):=\sum_{j=1}^\infty \exp(-a j^b ) , \quad a,b > 0 $$ which (due to monotonicity) can be bounded by $$ S(a,b) \leq \int_0^\infty \exp(-a x^b ) \, \mathrm{d} x = \frac{\Gamma(1/b)}{a^{...
5
votes
2answers
69 views

At what point does exponential growth dominate polynomial growth?

It's well-known that exponential growth eventually overtakes polynomial growth (link, link). So for any non-negative integer $d$ and positive $\epsilon$, there exists $t^* \ge 0$ for which $$ 1 + ...
3
votes
2answers
99 views

Solve $\sqrt x = 1 + \ln(3 + x)$ algebraically

I am having trouble with this homework problem. I am able to graph and find the solution, but I am curious as to how one would do this algebraically. The way I began, was subtracting $1$ on both ...
0
votes
4answers
116 views

Solving $e^x - 3 = 0$ [closed]

I want to solve this equation for $x$: $$e^x - 3 = 0$$ Can somebody give me some hints? Thanks.
0
votes
3answers
39 views

How do i solve these exponential equations? [closed]

Is there a way to solve these exponential equations without using logarithms? I tried to get the same base for all the terms, but I could not make it. Is there any other general procedure that I can ...
2
votes
0answers
55 views

Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
2
votes
4answers
77 views

How to solve an equation like $2{^x} + x = 2 $?

I encountered an equation similar to this in an old math exam. $2{^x} + x = 2 $ I couldn't figure it out and ended up with a mess of logarithmic functions. The answer sheet indicated it should be ...
0
votes
1answer
19 views

Differing graphs for simple inverse exponential problem

In class, we are learning exponential functions. The following inverse exponential problem is bothering me: $y=x^{-\frac{1}{9}}$. When graphed, I feel that it should look like it does on Desmos: ...
0
votes
1answer
32 views

How to solve the following limit using mathematics Stirling $\lim\limits_{n\to \infty}\frac{n!}{n^ne^{-n}\sqrt{2\pi n}}=1$

How to solve the following limit using mathematics Stirling $\lim\limits_{n\to \infty}\frac{n!}{n^ne^{-n}\sqrt{2\pi n}}$. a) $\lim\limits_{n\to \infty}\frac{n!e^n}{n^{n+1/2}}$. b) $\lim\limits_{n\to ...
7
votes
1answer
1k views

A curious property of $\operatorname{frac}(e\cdot k)$

Let $\alpha > 0$ be a real number and let us consider the set $S(\alpha)$ of those natural numbers $n$ such that the fractional part of $\alpha \cdot n$ "begins" with the representation of $n$ (in ...
-1
votes
2answers
48 views

Solving exponential-linear equation

How to solve the next exponential linear equation? $y=1.6^x+3x$ I need to find $x$ in terms of $y$
2
votes
3answers
67 views

On the integral of $e^{aix}$.

I've been trying a simple trick, but I'm unsure as to why it is failing. Let's say I wanted to compute $ \int^{\pi} _{ - \pi} e^{aix} dx$ for some constant a. Why won't this trick work? $ \int^{\pi} ...
4
votes
1answer
95 views

How do you prove $e^x=\exp x$ for real, non-rational $x$?

Let $\exp x=\sum_{n\geq 0} \frac {x^n}{n!}$. Let $e=\exp 1$. Let $a,x\in \Bbb R$, $a>0$. We define $a^x=\sup \{a^r:r\in \Bbb Q, r<x\}$. I've already proved that for $x=\frac pq \in \Bbb Q$, $...
4
votes
4answers
416 views

Uniqueness of exponential function

To my knowledge, the exponential function is the unique function satisfying $f'=f$ and $f(0)=1$ however, unless I've made a mistake, we have $$\frac{\partial}{\partial x} (ax)^x = x (ax)^{x-1} a = ...
2
votes
1answer
39 views

Exponential matrix decay

I’m working on contractive systems that have a system of ODE equations. I have an exponential matrix multiply by time that is for a given matrix $A$ I’m getting $e^{At}$. I want to know what are the ...
3
votes
2answers
76 views

What type of functional equation is this?

I'm trying to solve the following functional equation $f\left(x\right)=A\mbox{ exp}\left\{ \int\frac{1}{f\left(x\right)x^{2}+Bx}dx\right\}$ where $f\left(x\right):\mathbb{R}_{+}\rightarrow\mathbb{R}...
13
votes
3answers
147 views

How to express $f(n\alpha)$ in terms of $f(\alpha)$

Original question: Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\dfrac{a^x-a^{-x}}{2}$, where $a>0$ and $a\ne 1$, and $\alpha$ be a real number such that $f(\alpha)=1$. Find $f(2\...
3
votes
1answer
89 views

What is the evaluation of $\sum_{k=1}^{\infty}\frac{k^n}{k!}$? [duplicate]

I stumbled upon a similar problem and really liked the answers there, so I wondered if there were a general solution for $$\sum_{k=1}^{\infty}\frac{k^n}{k!}=?$$ Sadly, when I try to apply some of ...
1
vote
1answer
32 views

What function describes this problem of every possible breeding of a set of dogs?

If I have n dogs [a, b, c, ...], and I want to breed them in every possible combination (every possible binary tree made of ...
6
votes
0answers
108 views

An integral for $2\pi+e-9$

Motivation Lucian asked about the almost-integer $2\pi+e\approx9$ in a comment to a partially answered why question about $e\approx H_8$. This is more involved than approximations to $\pi$ and ...
1
vote
4answers
119 views

Show $\sum_{k=1}^\infty \frac{k^2}{k!} = 2\mathrm{e}$ [duplicate]

I stumpled upon the equation $$\sum_{k=1}^\infty \frac{k^2}{k!} = 2\mathrm{e}$$ and was just curious how to deduce the right hand side of the eqution - which identities could be of use here? Trying ...
1
vote
1answer
54 views

What are the solutions for $2^x=x^2$? [duplicate]

What are the solutions for $2^x=x^2$? I noticed there were 2 roots: $2,4$. Are there any other roots, and how do you calculate them?
0
votes
1answer
29 views

Solving an expression containing two added exponential functions

I have a problem solving the below equation with respect to $x$: $0.6\cdot \exp(\frac{-40}{x})+0.4 \cdot \exp(\frac{10}{x})=1$ My problem is that I have two exponential functions which are added ...
2
votes
3answers
99 views

Limit of $(1+3/n)^{4n}$ as $n$ goes to infinity

This afternoon I was trying to evaluate $$\lim\limits_{n \to \infty} \biggl( 1 + \frac{3}{n}\biggr)^{4n}$$ but was having some difficulty in doing so. I know the answer to be $e^{12}$, and can ...
0
votes
2answers
49 views

Simplify $\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$

How do you make the jump from: $$\frac{1-(\frac{4}{25})^{21}}{1-\frac{4}{25}}$$ To: $$\frac{25^{21}-4^{21}}{25^{21}-4(25^{20})}$$
4
votes
3answers
89 views

Prove Exponential Function Inequality: $e^x \le \frac{1}{1-x}$

Prove that $e^x \le \dfrac{1}{1-x}, x\lt 1.$ I find that if we set $f(x)=e^x(1-x)$ then $f(0)=1 $ and $f'(x)<0, x\in(0,1]$ proving the inequality for $x\in[0,1]$ but I don't see how to prove it ...
0
votes
3answers
47 views

how tell if a series of power numbers is bigger then others

I trying to order a list of mathematical expressions in string format as: "2*2" "4^1" "4^2^5" so far, so good for non exponential operations (^). I could compute ...
0
votes
1answer
67 views

How can one properly understand the fact that $e^x$ can be differentiated an infinite amount of times?

Simply put if I follow the rule derived by the simple proof denoting $e^x$ to be the derivative of $e^x$ then it follows that it should have an infinite number of derivatives. Is this a conceptual ...
11
votes
3answers
486 views

Number system with $e^x = 0$ for some $x$

It is well known that $e^x \ne 0$ for all $x \in \mathbb{R}$ as well as $x \in \mathbb{C}$. Upon reading this article and doing a bit of research I have found that this also applies to the ...
1
vote
1answer
31 views

Limit of a sequence of functions recursively defined by integrals

$f_n:[0,\infty)\to\mathbb{R}$ is defined recursively by $f_1:=0$ and $$f_{n+1}(x)=e^{-2x}+\int_0^xf_n(t)e^{-2t}dt,\qquad n\ge 1$$ I need to show that the limit $f(x):=\lim_{n\to\infty} f_n(x)$ exists ...
3
votes
3answers
77 views

Square root of $e^{ix}$

Is this $$\sqrt{e^{ix}}=e^{\frac{ix}{2}}=\cos\frac{x}{2}+i\sin\frac{x}{2}$$ true? Or $$\sqrt{e^{ix}}=\sqrt{\cos x+i\sin x}$$ How do I express square root of $e^{ix}$ as a non-square root expression?
7
votes
3answers
79 views

Limit similar to $\lim_{n \to \infty} \left(1-\frac{1}{n} \right)^n = \text{e}^{-1}$

I want to show that $$ \lim_{n \to \infty} \left(1-\frac{n}{n^2} \right) \left(1-\frac{n}{n^2-1} \right) \cdot \ldots \cdot \left(1-\frac{n}{n^2-n+1} \right) = \lim_{n \to \infty} \prod_{k=0}^{n-1} \...
2
votes
2answers
60 views

Proof that $\lim_{x\to\infty} b^x=0 \iff 0 \leq b<1$

Are there any errors in the following attempt to prove the above? $(\Leftarrow)$ Let $f(x)=b^x$, with $0 \leq b<1$. Then, for all $x$, $f(x)>0$ and $f'(x)=b^x \ln(b)<0$. This means that $f$ ...
1
vote
4answers
78 views

Complex Finite Product $\prod_{k=0}^{n-1} (1-\zeta^k z)$

I am working on a review for a graduate level Complex Analysis course. The following problem is on the review: Let $\zeta= e^{\frac{2\pi i}{n}}$ $(n\in \mathbb{N})$; show that $\displaystyle{\prod_{...