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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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Is there any formula or method to calculate logarithms with the following information available $y = a^{x}?$

I would like to know how calculators calculate log or how in ancient times, Mathematicians used to develop log sheets for specific variables. For instance, for $y = a^{x}$ we know that $$x =( y/a)-(a^...
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Does $180i = \pi(i)$ through euler's identity by $e^{90i} = 0 + i$

I have a strange question: If $e^{ix} = cosx + isinx$ then shouldn't $e^{90i} = cos90 + isin90$ Which should simplify to $e^{90i} = 0 + i$ making $e^{90i} = i$ But we already know that $e^{\pi i} = ...
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Intuitive way to understand the limit of $(1-\frac{1}{n})^{n}$ [closed]

I know that the limit of $(1-\frac{1}{n})^{n}$ is $\frac{1}{e}$ Please help me grasp why.. I also know that the limit of $(1+\frac{1}{n})^{n}$ is $e$.
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Implicit equation $\ln(\frac{x}{y})-y=1$ to rectangular equation not in terms of $W(x)$

Backstory and Other Info I'm not sure if this is possible, I'm currently a precalculus student and have a very limited understanding of much of any of this. However, I do like to go on WolframAlpha ...
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What is the limit of $(1+(\frac23)^n)^{1/n}$?

I was studying complex analysis and wanted to find the radius of convergence of the power series $$\sum_{n=1}^\infty\frac{2^n+3^n}{4^n+5^n}z^n$$ I used 'root test' and had to find the limit of the ...
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How to prove that Re(zeta(1+i/n)) is equal to the euler constant as n -> infinity

I have noticed that $\displaystyle\lim_{n \to \infty} \Re\left[\zeta\left(1 + \frac{i}{n}\right)\right] = \gamma$ How can this be proven?
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How to evaluate $\sum_{n=1}^{\infty}\frac{H_{kn}^2-[\gamma+\ln(kn)]^2}{n}?$

From this post @Olivier Oloa gives the closed form for this sum $(1)$ $$\sum_{n=1}^{\infty}\frac{H_n^2-(\gamma+\ln n)^2}{n}=\frac{5}{3}\zeta(3)-\frac{2}{3}\gamma^3-2\gamma\gamma_1-\gamma_2\tag1$$ I ...
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Calculation of Integrals with reciproce Logarithm, Euler's constant $\gamma=0.577…$

Evaluate the improper integral $\int\limits_0^1\left(\frac1{\log x} + \frac1{1-x}\right)^2 dx = \log2\pi - \frac12 = 0.33787...$ With integration by parts we get from $\int\limits_0^1\left(\frac1{\...
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Why did the natural constant e had to be 2.718.. [closed]

I tried hard to not make this duplicate but if I have missed it please flag this question. So people constantly ask and answer why e is very important, where it came from, and its derivations, etc. "...
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Prove $\gamma = \frac{1}{2} + 2 \cdot \int_0^\infty \frac{\sin(\arctan(x))}{(e^{2 \pi x} - 1) \cdot \sqrt{1 + x^2} } dx$

I've found the following integral on the Wikipediapage of the Euler-Mascheroni constant and I want to prove it. $\gamma = \frac{1}{2} + 2 \cdot \int_0^\infty \frac{\sin(\arctan(x))}{(e^{2 \pi x} - 1) ...
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Can someone shed some light on this inequality?

I have a question relating to image that I've attached. It is a proof that the sequence is increasing. I don't understand the logic behind the third equation $$\frac{a_{n+1}}{a_n}>\left (1-\frac{1}{...
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What is $\sinh(x)$? And $i\pi$

I've been looking around the internet for a way to solve $\sin(i)$, and I found something about $\sinh(x)$ I furthered my search and found something about $\sinh(x)= \frac{e^x - e^{-x}}{2}$. When I ...
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Computing the error function for Euler's number

By the error function for the sum $$\sum_{i = 0}^\infty \frac{1}{i!},$$ I mean the function $$f : \mathbb{R}_{> 0} \rightarrow \mathbb{N}$$ defined as follows. For each $\varepsilon \in \mathbb{R}...
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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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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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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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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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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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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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. 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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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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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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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
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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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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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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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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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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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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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1answer
141 views

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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67 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
301 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
40 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
59 views

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 ...