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

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2
votes
0answers
22 views

Exponential Growth/Compound Interest confusion?? [closed]

So I have this problem: $\$800$ is invested at a rate of $6.5\%$ and is compounded monthly for $5$ years. So I use the formula $A = 800(1 + 0.065/12)^{12 \cdot 5}$; however, I do not get the correct ...
5
votes
8answers
177 views

Prove: $(1-\frac{1}{d+1})^d>\frac{1}{e}$

I need to prove that $\left(1-\frac{1}{d+1}\right)^d>\frac{1}{e}$. I guess that I have to use that $\left(1+\frac{1}{n+1}\right)^n\rightarrow e$ for $n\rightarrow\infty$ or better $<e$ or ...
4
votes
2answers
251 views

If $x$ is rational, can $\log(1-x)/\log x$ be algebraic?

If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to an algebraic number, say $g$? $$\frac{\log(1-x)}{\log x} = g$$
3
votes
1answer
63 views

Why is the solution to $y' = y^n$ always in polynomial form EXCEPT when $n = 1$?

Could someone explain (intuition-wise) why the differential equation $$y' = y^n$$ for $n \in \mathbb{N}$ seems to always some kind of polynomial solution (or a ratio of polynomials, etc.) except ...
1
vote
3answers
37 views

Exponential distribution wait time probability

I would like to check my answer, I have been asked to work out the probability of value greater then 10 given an exponential distribution with a mean of 10. My intuition would be that this is equal ...
5
votes
7answers
155 views

Why does $e^{-(x^2/2)} \approx \cos[\frac{x}{\sqrt{n}}]^n$ hold for large $n$?

Why does this hold: $$ e^{-x^2/2} = \lim_{n \to \infty} \cos^n \left( \frac{x}{\sqrt{n}} \right) $$ I am not sure how to solve this using the limit theorem.
3
votes
3answers
61 views

Finding closed form expression for $ \sum_{n=1}^{\infty} n\left( e^{-n(n-1)} - e^{-n(n+1)}\right)$

I want to find a closed form expression for: $$ \sum_{n=1}^{\infty} n\left( e^{-n(n-1)} - e^{-n(n+1)}\right)$$ This isn't a homework problem, more of something that emerged as a side problem from ...
2
votes
3answers
50 views

Exponent $x$ tending to infinity

When an expression has $x$ tending to infinity and it's an exponent, like: $$\lim\limits_{x \to \infty} \frac 1{2^x},$$ is there some way to get a solid value for this limit (hopefully $0$, but at ...
1
vote
0answers
30 views

Exponential and Hyperbolic Functions Before Power Series

Is there a textbook that covers the exponential function and the possibly the hyperbolic functions from a precalculus geometrical viewpoint? I'm essentially asking for something like a trigonometry ...
0
votes
2answers
44 views

Closed form expression for Integral of exponential function

How to compute the following integral \begin{equation} \int_{n_0}^{\infty} \exp(- c x^\lambda) dx, \end{equation} when $0 < \lambda \leq 1.$
1
vote
2answers
35 views

Which is greater ? Sum of odd power terms or even power terms in the exponential Taylor series?

I came across this question, in a book. Define $f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{{(2n+1)}!} $ and $ g(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{{(2n)}!} $, where x is a real number. Then, ...
7
votes
2answers
85 views

Show that $p_n(a)\neq 0$ if $|a|=n$

I am working the next problem: Consider the polynomials $$ p_n(z)=\sum_{j=0}^{n}\frac{z^j}{j!} $$ For $n \geq 2$, show that if $a \in \mathbb{C}$ is such that $|a|=1$ or $|a|=n$, then ...
1
vote
1answer
278 views

Limit involving tetration

Let the notation be $a^{\wedge\wedge}b = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{b\,times}$ for tetration. My mentor conjectured the following: Let $n$ be a positive integer, then let $A(n)$ be ...
1
vote
2answers
38 views

Link between exponential distribution and poisson probability mass function

Customers arrive randomly and independently at a service window, and the time between arrivals has an exponential distribution with a mean of 12 minutes. Let X equal the number of arrivals per hour. ...
0
votes
0answers
51 views

Normal Exponential Convolution proof

I am seeing a scientific paper where they explain background correction modelled as two random independent variables one with exponential distribution and the other with normal distribution. ...
4
votes
5answers
103 views

Derivation of the Exponential Nature of $e^x$

Presumably, the transcendental number $e$ was first found by taking the power series solution to the (arguably most fundamental) differential equation $f'(x)=f(x)$, with the initial condition $f(0)=1$ ...
0
votes
6answers
43 views

How to prove an exponential equation.

Is there any law of exponents that applies to this equation? How can show that the LHS gets converted into the RHS $(e^x −e^{−x})^2 =(e^x +e^{−x})^2 −4$
0
votes
2answers
46 views

Inverse of a function $xe^x$

How should I proceed about finding the inverse of the function $xe^x$? I have been wondering about it for a long time and can't think of anything to do.
1
vote
0answers
114 views

Integral of $xe^{-ax^2-bx^{-1}}$

I am currently facing an integral I have no clue how to solve it. I believe it is rather exoctic, but I hope you might have some good advice: $$\int_0^{\infty} x e^{-ax^2-bx^{-1}} \, \mathrm{d}x$$ ...
6
votes
1answer
62 views

Is there a constructive discontinuous exponential function? [duplicate]

It is well-known that the only continuous functions $f\colon\mathbb R\to\mathbb R^+$ satisfying $f(x+y)=f(x)f(y)$ for all $x,y\in\mathbb R$ are the familiar exponential functions. (Prove ...
0
votes
0answers
16 views

Optimal Space-Travel Departure Time (Issues deriving and solving complex expressions).

Problem This problem aims to determine the optimal time to depart for an intergalactic destination, taking into account the fact that in a number of years technology back on the planet you left may ...
3
votes
2answers
44 views

How to precisely define $C^\infty$ in $f(x) \in C^\infty$

In single variable calculus, a common way to denote a function that is continuous for all derivatives is to write $f(x) \in C^\infty$ i.e. $f(x) = \exp(x)$ Is there a more rigorous way to define ...
0
votes
1answer
35 views

Exponential type of $\sin z$

An entire function $f$ is of exponential type if $\,\lvert\, f(z)\rvert\le C\mathrm{e}^{\tau\lvert z\rvert},\,$ for all sufficiently large values of $\lvert z\rvert$. The exponential type of $f$ is ...
0
votes
0answers
12 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
1
vote
1answer
35 views

Translate exponential distribution into normal distribution

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the ...
2
votes
2answers
77 views

Solve matrix equation $e^A=e^B$ for nilpotent $A, B$.

I need to solve equaton $e^A=e^B$ for nilpotent matrices A and B over field $\mathbb C$, where $B$ is fixed. I solved equation $e^X=E$ for all matrices. The solution is any semisimple (in case ...
6
votes
2answers
76 views

Can I interpret the exponential of the derivative operator, $e^D$, as infinite shift operators each shifting “infinitesimally”?

To better explain what I mean, an example can be very useful. Consider $e^{i\theta}$. We could express this using the series definition or the limit definition of $e^x$ instead: $$e^{i\theta} = ...
6
votes
1answer
124 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
2
votes
0answers
27 views

Maximum density linear combination chi squares

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i) \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound ...
0
votes
1answer
20 views

Values of $p$ for which equation $p3^x+2\cdot 3^{-x}=1$ has a unique solution

$p3^x+2\cdot 3^{-x}=1$ I got this down to a quadratic equation by marking $3^x$ as $t$ and I fiddled with the stuff and got some solutions that apparently don't fit the real one in the textbook was. ...
1
vote
0answers
20 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
5
votes
3answers
122 views

How can I calculate $\lim_{n \to \infty} (1 + \frac{1}{n!})^n$ and $\lim_{n \to \infty} (1 + \frac{1}{n!})^{n^n}$?

How do you calculate the following limits? $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^n$$ $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^{n^n}.$$ I really don't have any clue ...
3
votes
2answers
95 views

Showing $r\in\mathbb Q\setminus \{0\}\implies e^r\notin \mathbb Q$

For a given $n>0$, let $\displaystyle J_n:x\to \frac{1}{n!}\int_{-x}^x(x^2-t^2)^ne^tdt$ a. Prove that there exists $A_n,B_n\in \mathbb R_n[X]$ such that $\forall x\in \mathbb R^+, ...
6
votes
3answers
207 views

What is the $\lim_{n\rightarrow \infty }(1+\frac{1}{n})^{n^n}$

What is the $$\lim_{n\rightarrow \infty }\left(1+\frac{1}{n}\right)^{n^n}$$ I know that the $\lim_{n\rightarrow \infty }\left(1+\frac{1}{n}\right)^{n}=e$, so I wanted to find the limit by the same ...
0
votes
0answers
41 views

Algebra Integral simplification

Let some equation problem final result is like this $0\leq1\leq s\leq t\leq u\leq v$ \begin{align} M=\mathrm{exp}\bigg\{-\pi\lambda v^2+\pi\lambda v^2\bigg(\displaystyle\int_o^s ...
1
vote
3answers
50 views

Explicit form of this series expansion?

I am considering the following series expansion: $$f(k):=\sum_{n\geq 1} e^{-k n^2}$$ with $k>0$ a fixed parameter. Is there a possibility to either find a closed form expression for $f(k)$? Or at ...
2
votes
0answers
28 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))] = ...
2
votes
0answers
37 views

Invertibility of infinite order matrix

how the matrix $[e^{-(x_j-x_k)^2}]$ is invertible where $\{x_j\}$ be any real sequence such that $(x_{j+1}-x_j) >0$ for all $j \in Z$ where $Z$ denotes the set of integers.
2
votes
2answers
31 views

Tweaking formulas to increase scoring

I am building a model for SVM classification. However, the confidence score that i have would be from negative to positive. This is the formula i am using to normalize the confidence score ...
0
votes
0answers
16 views

Finding probability of being in a certain state in a CTMC.

There are two transatlantic cables each of which can handle one telegraph message at a time. The time to breakdown for each has the same exponential distribution with parameter λ. The time to repair ...
5
votes
6answers
195 views

Are the any **non-trivial** functions where $f(x)=f'(x)$ not of the form $Ae^x$

This may seem like a silly question, but I just wanted to check. I know there are proofs that if $f(x)=f'(x)$ then $f(x)=Ae^x$. But can we 'invent' another function that obeys $f(x)=f'(x)$ which is ...
3
votes
0answers
55 views

What is $\int \frac{e^{a x}}{1+x^2} dx $?

In my answer to another question (here: Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$), I came up with this integral: $\int \frac{e^{a x}}{1+x^2} dx $. I ...
3
votes
3answers
93 views

Apex of an Exponential Function

Is there a way of calculating where the apex of an exponential lies? There's probably a deeper / more mathematical way of explaining what this is exactly. The image hopefully demonstrates what I mean. ...
0
votes
3answers
28 views

Show that $ f(x) = A.exp(2x) $ if $ f'(x) = 2f(x) $

Show that $ f(x) = A.exp(2x) $ if $ f'(x) = 2f(x) $ for some $ A \in \mathbb{R} $ Is it sufficient to say that if the derivative of a function contains itself, then it must be the exponential ...
3
votes
1answer
64 views

Evaluate the limit $\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $

It seems reasonable to assume that $$\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $$ goes to zero but I can't figure out how to prove it.
1
vote
1answer
28 views

Determine the order of $\exp Q(z)$ when $Q$ is a polynomial of degree $q$.

I am looking to determine the order of $f(z) = \exp Q(z)$ when $Q$ is a polynomial of degree $q$. I think the order is $q$, but I am struggling to prove it. The definition of order is: An entire ...
0
votes
2answers
52 views

L'Hôpital's rule exercise with sqrt(x) as exponent

I'm a bit stuck trying to find the limit of the following function: $$\lim_{x \to 0^+}\,\,{x^{\sqrt{x}}} $$ We are expected to use L'Hôpital's rule, and thus far I've managed to resolve the equation ...
2
votes
2answers
50 views

Proving $\log n < \sqrt n$

I am trying to prove $\exists n_0 > 0: \forall n > n_0: \log n < \sqrt n$. My attempt uses the series representation of the exponential function, but it does not seem to accomplish the proof: ...
0
votes
1answer
11 views

Distribution of exponential(X/c)

Suppose $X \sim Exponential(\lambda)$. That is, the PDF for $X$ is $f_X(x)=\lambda \cdot e^{-\lambda x}$, $x\ge 0$, and the CDF of $X$ is $F_X (x)=\int_{-\infty}^x f_X(x)=1-e^{-\lambda x}$, $x\ge ...
0
votes
0answers
46 views

Solving a ratio of summation

I have to solve the equation \begin{equation*} y = \frac{\sum_{j=1}^m a_j x'_j}{\sum_{j=1}^m a_j x_j} \end{equation*} We have $\sum_{j=1}^m a_j \frac{x'_j}{x_j} = 1$ and $\sum_{j=1}^m a_j = 1$ ...