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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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Are there any approximations of Euler's Constant? [on hold]

Some math person in Instagram posted the following... $$ e =\frac {200\sqrt{3} \pi^2 +199\sqrt{3} \pi - 4}{400\pi + 398} $$ claiming it is of their own finding. Is this true? if not..is there ...
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1answer
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Prove $\sum_{n=1}^\infty \frac{\zeta(2n+1) - 1}{(2n+1) 2^{2n}} = 1 + \ln(2) - \ln(3) - \gamma$

I've found the following series on the Wikipediapage of the Euler-Mascheroni constant and I want to prove it. $$\sum_{n=1}^\infty \frac{\zeta(2n+1) - 1}{(2n+1) 2^{2n}} = 1 + \ln(2) - \ln(3) - \gamma$$...
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1answer
37 views

Simplify $\lim_{n \to \infty}\frac{(1-A)(1-B)}{(2-A-B)(A-B)}=\frac{\gamma(\gamma-1)}{2\gamma-1}?$

Let: $$A=\Gamma\left(1+\frac{1}{\Gamma\left(\frac{1}{n}\right)}\right)$$ $$B=\Gamma\left(1+\frac{1}{1+\frac{1}{\Gamma\left(\frac{1}{n}\right)}}\right)$$ I spent a few days trying to work on the $\...
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1answer
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Do the Prime Number Theorem and/or Riemann Hypothesis predict a limit on the accuracy of this formula for $\gamma$?

This question is related to the following formula for Euler's constant $\gamma$ where $A$ is Glaisher's constant. (1) $\quad\gamma=12\,\log(A)-\frac{\pi^2}{6}\sum\limits_{n=1}^N\frac{\mu(n)}{n^2}\,\...
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1answer
35 views

Contradiction in Euler’s Identity? $2i\pi = 0$ [duplicate]

Many of us know Euler’s famous identity: $e^{i\pi} + 1 = 0$. But if we add -1 (subtract 1) to both sides we get: $e^{i\pi} = -1$ then natural log of both sides and: $i\pi = \ln(-1)$ Next ...
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Notation $\gamma$ for Euler's constant $\gamma$.

Question. In which book or article George Boole used the notation $\gamma$ for Euler's constant? Background. Today, Euler's constant is usually denoted by $\gamma$. In 1993 it was found out that the ...
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0answers
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Table of integrals with Euler's constant gamma.

I'm working on a large table of integrals with Euler's constant $\gamma=0.577...$ It's a very interesting point, that a lot of integrals are found in similar tables like Gradshteyn-Ryzhik, but are not ...
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1answer
43 views

Use Fermat’s Theorem to prove Euler’s Theorem in the case m = pq. with p and q being two distinct prime numbers

If $p$ is a prime and $p$ does not divide $a$, then $$a^{p-1} \equiv 1 \pmod{p}.$$ Since $p$ is prime, the fact that $p$ does not divide a means that $a$ and $p$ are relatively prime. Also, $\varphi(p)...
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Integral of the difference between a function and its floor

I know the following identity: $$\int_1^\infty\left(-\frac{1}{x}+\frac{1}{\lfloor x\rfloor}\right)dx=\gamma$$ I have a function $f(b)=0.5\left(\sqrt{(N-b^2)}-b+1\right)$. Can a similar identity be ...
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1answer
38 views

. Suppose gcd(a, m) = d and that m > 1. Consider the congruence ax ≡ b (mod m). Should there be a solution for every choice of b?

If yes, prove your claim; if not, give a counter example X is not told so I assume it can be any orbitrary number. b is also abitrary, with that being said, isn't true due to those 2 factors?
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23 views

Limit inside the exp function

I do not understand the following limit-rule: $$\lim_{x\to\infty}\exp(f(x))=\exp(\lim_{x\to\infty}f(x))$$ Why is that true?
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1answer
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Unicode notation of Euler Constant [closed]

According to Wikipedia, eulers-constant is usually denoted as $\gamma$. Why does Unicode use 'ℇ' to denote this number? Wikipedia redirects 'ℇ' to eulers-constant, but in that page, there's no ...
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1answer
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Clever equalities proven similarly to Euler's Identity

From How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?, a very elegant proof of Euler's Identity was given. Namely, observing $f(z)=g(z)h(z)=e^{-iz}(\cos(z)+i\sin(z))$, ...
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What mathematical consequences might there be if Euler Mascheroni constant is rational?

So far as I know, no one has proved the irrationality of Euler Mascheroni constant. There are discussions about the difficulty of proving the irrationality of this constant. Since we cannot prove ...
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1answer
64 views

I need to calculate a limit using Euler number [closed]

How would one evaluate this limit without using L'Hôpital's rule? What I know for sure is that this limit equals to zero, but I don’t know how to solve it. $$ \lim_{x\rightarrow \infty}\left(\frac{2x-...
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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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1answer
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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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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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1answer
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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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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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5answers
286 views

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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1answer
58 views

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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2answers
156 views

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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1answer
64 views

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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1answer
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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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1answer
98 views

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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1answer
89 views

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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1answer
129 views

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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2answers
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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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2answers
29 views

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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1answer
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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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3answers
64 views

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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2answers
254 views

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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1answer
39 views

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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3answers
174 views

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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2answers
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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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3answers
48 views

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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2answers
64 views

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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170 views

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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1answer
53 views

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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3answers
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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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1answer
75 views

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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2answers
69 views

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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2answers
68 views

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 ...
1
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5answers
103 views

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 ...
0
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3answers
294 views

$\ln(x) -\ln(y) = kt$, 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 $x = ye^{kt}$. ...
2
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2answers
153 views

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 ...