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Questions tagged [eulers-constant]

For questions related to Euler's constant $\gamma$, which is defined to be the limiting difference between the natural logarithm and the harmonic series.

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How can this Infinite series be proven? [duplicate]

Recently i read that ( i cannot remember where though) $$\sum_{k=0}^{\infty}\frac{1}{(2k)!!} = \sqrt{e} $$ Wolfram also confirms this , but I have been looking everywhere to find a proof and i ...
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I have found a way of computing Euler's number. Is there any possible intuition of how that might be the case?

So a few days ago I just kind of messed around with my calculator, when I had an idea about a new continued fraction. I inputted it, and I found that it converged really quickly, and, quite wondrously,...
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Please help explain the maths of this diffusion model

Diffusion model Could anyone help with my understanding of the maths in this excerpt from a physiology textbook (please see link). The authors describe a model of gas diffusion across the length of ...
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Integral with Euler-Mascheroni constant

I want to solve the following integral $$\int_0^\infty \frac{\sin{x}\ln{x}}{x}dx$$ To make things worse (or better?), I parameterize the sine function to $e^{-ax}$. \begin{align*} I(a)&=\int_{0}^\...
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1answer
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Why is $\cos(i)-i \sin(i)=e$? [duplicate]

When I was typing $\cos(i)-i \sin(i)$ into the calculator, I found out that it is equal to e (Euler's Constant). I was amazed by that "discovery" so I checked in on the internet and there was no ...
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A nice relationship between $\zeta$, $\pi$ and $e$

I just happened to see this equation today, any suggestions on how to prove it? $$\sum_{n=1}^\infty{\frac{\zeta(2n)}{n(2n+1)4^n}}=\log{\frac{\pi}{e}}$$
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A question about exponential functions

How can we prove that for b>a>e (e being the euler’s number), a^b is greater than b^a?
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Continued fractions approximation using golden ratio

Hello today my friend helped me with my problem, but he did not give me any additional informations why it works like that. Let's suppose that I need to get ln(n) using continued fractions. He told ...
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Is there a proof of an infinite number of prime numbers using the irrationality of $e$?

That the set of prime integers is infinite can be proved using the irrationality of $\pi$; see this wikipedia link. It analyzes the representation $\tag 1 {\displaystyle {\frac {\pi }{4}}={\frac {3}{...
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Approaching The Euler-Mascheroni Constant

I am looking for a value $a \approx 14$ with some nice property. So I am going to define some things with this value $a$ and then ask what $a$ does the trick I want (If there is some $a$ that does the ...
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$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!} = \frac{1}{2}$ - basic methods

Prove that $$\lim\limits_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!} = \frac{1}{2}$$ This problem appeared on MSE many times, but each time it was solved using Poisson distribution or lots of ...
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description in mathematics [closed]

I have a general question. I would like to describe the different ways of representing euler's number. My question is how to describe something in mathematics. The Euler number can be represented ...
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the basis of the natural logarithm

Euler found out that Euler's number was the basis of the natural logarithm. He started with this equation for infinitely small numbers $w$: For $a>1$: $$a^w=1+kw$$ $w$=infinitely small numbers $...
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Are there any theorem relating transcendency/irrationality of Euler's constant and the Riemann hypothesis?

Several formulations of the Riemann hypothesis can be given in terms of Euler’s constant. For example, Theorem (Nicolas 1981) The Riemann hypothesis holds if and only if all the primorial numbers $...
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Evaluation of $\sum_{n\geq1}\frac{1}{n}\ln(1+\frac{1}{n})$

Coming across the calculation of a special integral I get stuck on the following series, which I have given its integral representation : $$\text{J}=\sum_{n\geq1}\frac{1}{n}\ln\bigg(1+\frac{1}{n}\...
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Evaluation of $\int_{0}^{1}\int_{0}^{1}\{\frac{1}{\,x}\}\{\frac{1}{x\,y}\}dx\,dy\,$

Let $\{\}$ denote the fractional part function, does the following double integral have a closed-form ? $$\int_{0}^{1}\int_{0}^{1}\bigg\{\frac{1}{\,x}\bigg\}\bigg\{\frac{1}{x\,y}\bigg\}dx\,dy\,$$
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How can one evaluate the following series:

$$\sum_{k=1}^{\infty} (-1)^{k}\frac{(\ln{k})^{2}}{k} \space\space\space ?$$ From $\sum_{n\geq 1}\frac{(-1)^n \ln n}{n}$ it has been answered that $$ \sum_{k=1}^{\infty} (-1)^{k} \frac{\ln{k}}{k} = \...
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Fractional part of the floor function Integral [closed]

Let $\lfloor\rfloor\ $ and $\{\}$ denote the floor function and the fractional part funtion, respectively. Then calculate in closed-form the following integral $$\int_{0}^{1}\bigg\{\frac{1}{x}\bigg\...
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Evaluation of a series involving the harmonic number

Let $\text{H}_n$ denote the n-th harmonic number, does the following sequence have a closed-form ? As an approximation I have got $0.0922514...$ $$ \frac{3}{2}+\lim_{n\to\infty} \bigg(\sum_{k=3}^{n}\...
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Square root of fractional part integral

Does the following integral have a closed form ? $$\int_{0}^{1}\sqrt{\bigg\{\frac{1}{x}\bigg\}}dx$$ Where $\{x\}$ denotes the fractional part of $x$.
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Nontrivial solutions to $\sum f(x) - \int f(x) = \gamma $?

Consider $$ T(f) = \lim_{t \to \infty} \sum_{i =1}^t f(i) - \int_1^t f(x) dx $$ Notice $ T(a + b) = T(a) + T(b) $ and $T(C a) = C T(a)$ when $C $ is a constant. Also $T(0)= 0$. Let $T(d) = 0$. So $...
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Irrationality of the Euler–Mascheroni constant

The definition of the Euler–Mascheroni constant is the limit of $$H_n - \log(n)$$ as n approaches infinity. So, why is it so hard to prove the irrationality of this constant? $H_n$ is defined only for ...
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Check proof of $\int_{0}^{\infty}\sin(x\sinh\alpha)\sin(x\cosh\alpha)\ln(x)\frac{dx}{x}=-\alpha\gamma$

The question of the proof of this identity was raised today by user567627, but unfortunately the question was deleted by "community" without providing a reason. My answer (which had already received ...
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On the series $\sum_{n=1}^{\infty} \frac{\left \lfloor \log_2 n \right \rfloor}{2n(2n+1)(2n+2)}$ [closed]

How would one go to prove that $$\gamma = \frac{1}{2} + \sum_{n=1}^{\infty} \frac{\left \lfloor \log_2 n \right \rfloor}{2n(2n+1)(2n+2)}$$ where $\gamma$ stands for the Euler - Mascheroni constant ...
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How is $1- \frac{1}{2} +\frac{1}{3}…$ till infinity $= 2\log_e2$?

How is $1- \frac{1}{2} +\frac{1}{3}...$ till infinity $= \log_e2$? I know that $e^x = 1 + \frac{1}{1!} + \frac{1}{2!}...$ but how is this result gotten? I don't have a very extensive knowledge of ...
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How is $ \sum [(2/(n-1)!) + (1/n!)] = 2e + e - 1$?

A step in the solution of a question I'm doing says that $ \sum [(2/(n-1)!) + (1/n!)] = 2e + e - 1$. How is this possible? Isn't e equal to $ 1/0! + 1/1! + 1/2!... $ till infinity? Also how does -1 ...
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Solving $e^{\sin x} + e^ {\cos x} = e + 1$

How can we solve this equation? $$e^{\sin x} + e^ {\cos x} = e + 1$$ I know that $x=0$ and $x=\pi/2$ are solutions, as well as their periods, and that $x$ has to be between $0$ and $\pi/2$, but I ...
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Number of real roots of equation $e^{\cos x} - e^{-\cos x} - 4 = 0$?

I don't have much knowledge of Euler's number. How do I approach this problem? I tried taking the logarithm of both sides and ended up with $$ \log_e e^{2 \cos x} = \log_e 4 $$ giving $$ 2 \cos x = \...
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Is there another proof for Euler–Mascheroni Constant?

Problem Prove that the sequence $$x_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\ln n,~~~(n=1,2,\cdots)$$is convergent. One Proof This proof is based on the following inequality $$\frac{1}{n+...
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Why does the sign flip when substituting in Euler's formula

Why does the sign flip when substituting in Euler's formula into the below \begin{eqnarray} \lambda - 2 + \lambda^{-1} &=& C^2_x(e^{ip} - 2 + e^{-ip}) + C^2_y(e^{iq} - 2 + e^{-iq})\\ &=&...
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Why does the constant of integration move?

I'm trying to make sense of a solution to a separation of variables question. Namely where it goes from: $$y = \exp(-\cos x + C)$$ To: $$y = A\cdot \exp(-\cos x)$$ I understand the constant of ...
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What are the roots of the equation $z^{40} - z^{20} - a(a+1)$?

I found this question in a book. The answer given is that the roots are $(a+1)^{1/20}\exp\left({\frac{i2k_1\pi}{20}}\right)$ and $(a+1)^{1/20}\exp\left({\frac{i(2k_1+1)\pi}{20}}\right)$. How do I ...
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What's the process in rectangular form for deriving $(a+bi)^{a+bi}$?

What's the process, using Euler's Formula, solving $(a+bi)^{a+bi}$ when outlined algebraically in rectangular form? Edit: And also solving this in the same form and process, $(d cos(y)+i d sin(y))^{(...
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Is $\lim_{x\to \infty}{\sum_{i,h=1}^x \frac{1}{i^h} - x-\ln{x}}$ equal to $\gamma$, the Euler-Mascheroni constant? If so, why?

I was looking at the sum $\sum_{i,h=1}^x \frac{1}{i^h}$ on Desmos, and I realized it seemed to converge to the line $y=x$. When I subtracted x from it and increased the bounds, it seemed to be ...
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Explicit series for the minimum point of the Gamma function?

Is there any explicit series, product, integral, continued fraction or other kind of expression for the point at which $\Gamma(x)$ has a minimum in $(0,1)$? The decimal value can be found here http:/...
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Intuitive understanding of some important exponential functions

After watching these 2 brilliant YouTube Videos-- 1) Euler's formula with introductory group theory 2) But what is the Fourier Transform? A visual introduction. I am facing some difficulty in ...
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[[complex]] Solutions of $x=\sqrt{\sin{x}}$

I wanted to make one of those cool infinite recursive definitions for myself, and I chose one that I thought looked cool: $x=\sqrt{\sin{x}}=\sqrt{\sin{\sqrt{\sin{\sqrt{\sin{...}}}}}}$ for no other ...
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Can this be correct $\lim_{n\to \infty}\int_0^{\pi\over 2}\cos x\cos\left({x\over n }\right)\ln(\ln\csc x) \, dx = -\gamma?$

$$\lim_{n\to \infty}\int_0^{\pi\over 2}\cos x\cos\left({x\over n } \right) \log(\log\csc (x)) \, dx$$ surprisingly give a result $-0.577156\ldots$, which went we check up it, it is Euler's constant ...
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Help me understand e (exponential/euler's number) and how taking e to a certain power changes time period and return??

Ok, so I understand $e$ is just $\lim\limits_{n\to \infty}(1 + \frac{1}{n})^n$, and ends up being $2.71828\ldots$ It assumes the annual return you are getting is $100\%$ and is continuously compounded....
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How to find $x$ in $\sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{\infty}x^{ij}=\frac{1}{1-\gamma}$ if $0\lt x\lt1$? [closed]

I calculated the approximate sums and it seemed to be the case that $x=\gamma$ so far I cannot prove it.
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Is there a name for the theorem that $\lim\limits_{n \to \infty} (1+\frac{1}{n})^n < \infty$?

Is there a name for the theorem that $\displaystyle \lim_{n \to \infty} (1+\frac{1}{n})^n < \infty$ ? Wikipedia has a List of things named after Leonhard Euler which mentions Euler's number but not ...
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Prove that $\frac{x}{e^x}$ tends to zero as $x \to \infty $ [duplicate]

As the title states, I want to prove $$\lim_{x \to \infty} \frac{x}{e^x} =0$$ Clearly, L'Hopital's rule easily solves this. However, I'm curious to see if there's another way to prove it, without ...
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$\ln(x) -\ln(y) = kt$, make x the subject.

So I thought you take the inverse function of the whole expression getting: $x - y = e^{kt}$ and so your final answer would be $x = e^{kt} + y$ but according to the answers in the book $x = ye^{kt}$. ...
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What does ln() accomplish on a regression input?

I have gotten interested in forecasting using linear/nonlinear regression, particularly using Facebook's Prophet library for R/Python. It makes forecasting on a time-series input pretty ...
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4answers
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Why is $e^{i\pi}= -1$ [closed]

Why does $ e^{i\pi} = -1 $ ? I know that this form can be used to for instance act on a bloch's sphere (quantum mechanics) using it as $ e^{i\pi/4} $ will do a $ \frac{\pi}{4} $ rotation on the $x-...
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How to calculate the error of the series that originates the Euler's constant

I have been asked to calculate how many terms of the series that defines the Euler's constsant $\gamma$ to add at least, to calculate the value of $\gamma$ with error less or equal to $3 \times 10^{-3}...
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2answers
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Trying to solve a demonstration with the constant of Euler

I think this will be an easy problem for you, but I do not see the solution. I know that $$\displaystyle\sum_{k=0}^{\infty} \frac{1}{k!}=e$$ Knowing this, how can I demonstrate this $$\...
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Divergence of $\sum_{n=1}^\infty\frac{\mu(n)}{\sqrt{n}}\cos\left(n^2 \pi \gamma\right)$, where $\gamma$ is the Euler-Mascheroni constant

We denote the Möbius function as $\mu(n)$, see its definition from this MathWorld. On the other hand it is not known if the Euler-Mascheroni constant is irrational. After I've read a MathOverflow ...
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What is the series representation of $\frac{1}{\gamma}$?

If a series representation of ${\gamma}$ The Euler-Mascheroni constant is:$\displaystyle\sum_{k=1}^{+\infty}\left[\frac 1 k -\log\left(1+\frac 1 k\right)\right]$ then what is the series representation ...
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Raising a logarithmic function by e

For example if we have $$ \ln{y}=a\ln{x}$$ If we raise both sides to the power of e: $$ y = e^a .e^{\ln{x}} = e^ax$$ However by using log rules we get a different solution i.e. by letting $$a\ln{x}...