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

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4
votes
1answer
73 views

Intuitive reason for why the Gaussian integral converges to the square root of pi?

This is a very famous problem, which is commonly taught when students begin learning about multivariable integration in polar coordinates. However, it has always bothered me that we recieved such an ...
30
votes
2answers
654 views

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\left\{\frac{1}{x_{1}\cdots x_{n}}\right\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_{0}^{1} \! \cdots \! \int_{0}^{1} \left\{\frac{1}{x_{1}x_{2} \cdots ...
2
votes
1answer
59 views

Asymptotics of coefficients $[x^n] \frac{1}{\Gamma(1+x)}$ as $n$ is great

I am interested in the behaviour, as $n$ is great, of the coefficients $g_n$ in the Maclauren expansion of $\displaystyle \frac{1}{\Gamma(1+x)} $. We have $$ \frac{1}{\Gamma(1+x)}=\sum_{n=0}^\infty ...
9
votes
1answer
141 views

An integral with $e^{1+e^x}$ I had trouble working through

I had an analysis test earlier this morning and came across this integral, which I couldn't figure out. Parts of it are easy, but after integrating $y$ you're left integrating $xe^{1+e^x}$ which had ...
1
vote
1answer
79 views

An integral representation for $\psi$

Let $\displaystyle \gamma$ denote the Euler constant defined by $\displaystyle \gamma := \lim\limits_{n \to \infty} \left(\frac11+\frac12+\cdots+\frac1n- \log n\right)$. Here is an integral for ...
5
votes
1answer
160 views

A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form for the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = ...
2
votes
1answer
603 views

Simple proof Euler–Mascheroni $\gamma$ constant

I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \sum\nolimits_{k=1}^n \frac{1}{k} - \log(n)$ converges. At first I want to know if my answer is OK. ...
5
votes
2answers
116 views

Integral $\int_0^{\infty} \log(x) e^{-x^2} \mathrm{d}x = -\frac{1}{4}\sqrt{\pi} (\gamma + \log(4)).$

While trying to compute the expected value $E[\log(X)]$ for a normally distributed variable $X$ I found the following integral $$\int_{0}^{\infty}\log\left(x\right) {\rm e}^{-x^{2}}\,{\rm d}x =-\,{1 ...
0
votes
0answers
35 views

Step in proof involving Euler-Mascheroni constant

I was just looking at a proof that shows how the Euler-Mascheroni constant exists and is situated between 0 and 1 . However, I stumbled across a step in the proof that doesn't seem very obvious to me ...
0
votes
2answers
63 views

For which positive numbers $a$ is it true that $a^x \ge 1 + x$ for all $x$?

For which positive numbers $a$ is it true that $a^x \ge 1 + x$ for all $x$? This is question 24, chapter 3 from Stewart's Early Transcendental problems plus. I think I'm on the right track, but I'm ...
0
votes
1answer
43 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
2
votes
2answers
34 views

Zeta Function $\zeta(1\pm1/n)$ and Euler's constant.

How do I show that $$\lim_{n\to\infty}{\zeta(1+1/n)+\zeta(1-1/n)}=2\gamma$$ and $$\lim_{n\to\infty}{\zeta(1+1/n)-\zeta(1-1/n)}=\infty,$$ where $\gamma$ is the Euler's constant?
2
votes
2answers
158 views

Why is $1 + \frac{1}{2} + \frac{1}{3} + … + \frac{1}{n} \approx \ln(n) + \gamma$?

On StackExchange, I read that the harmonic series up to $\frac{1}{n}$ is approximately $\ln(n) + \gamma$, where $\gamma$ is the Euler-Mascheroni constant, which is close to $0.5772$. When I researched ...
2
votes
4answers
181 views

Compute $\sum (1+\frac12+\dotsb+\frac1n-\ln (n+\frac12)-\gamma)$

Compute $$\sum_{n=1}^\infty \Big(1+\frac12+\dotsb+\frac1n-\ln (n+\frac12)-\gamma\Big)$$ where $\gamma$ is Euler's constant It seems to be difficult, I have no idea go get started Thank you very ...
3
votes
2answers
65 views

Is this a legal transformation?

To be found: $$\lim \left(1+\frac{2}{n}\right)^n$$ Presuppose $~~\lim \left(1+\frac{1}{n}\right)^n=e~~$ is already shown. Expanding the first equation: $$\lim ...
0
votes
0answers
54 views

Euler's method and Kutta's [duplicate]

dy/dx=xy2 given $y(0)=1$ (i) Find the analytical solution to this problem. (ii) Given that $y(1.4)=50$, use the modified Euler method to calculate $y(1.5)$. Take $h=0.1$ and work to 4 decimal place ...
0
votes
1answer
28 views

Upper bound of natural logarithm

I was playing looking for a good upper bound of natural logarithm and I found that $$\ln x \le x^{1/e}$$ apparently works: Can someone give me a formal proof of this inequality?
1
vote
1answer
50 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
1
vote
2answers
73 views

Limit of ${x}^{\frac1{1-x}}$ as $x$ approaches 1

I have the following homework question, and I've been stuck on it for the last 2 hours. I have no idea how to find the answer. $$\lim\limits_{x\to1} {x}^{\frac1{1-x}}$$ I have tried manipulating the ...
11
votes
1answer
557 views

Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches ...
0
votes
2answers
49 views

How to isolate a variable when its on both sides of equation

$ ex-1=x $ How do rearrange this equation to isolate x i.e in the form of x=.....
3
votes
1answer
69 views

Why is $\int |e^{ix}|^2 dx = x + C$?

Quick question: Wolfram Alpha tells me that $$\int |e^{ix}|^2 dx = x + C$$ Why is that?
1
vote
0answers
30 views

Mertens Constant and Euler–Mascheroni constant

I found this titillating equation: $$M = \gamma + \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right]$$ where $\displaystyle M=\lim_{n \rightarrow \infty } \left( \sum_{p\,\leq ...
2
votes
1answer
71 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
0
votes
1answer
46 views

Dynamic Sizing of Circles Along a Logarithmic Spiral

I have created an logarithmic spiral in HTML canvas, and plotted circles along it. Using your mouse scroll wheel you can zoom in and out of the spiral (which works) – but I am having problems updating ...
3
votes
3answers
103 views

definition of the constant $e$

To my knowledge there are two possible ways to define $e^x$ $$e^x = \sum_{i=0}^{\infty}\frac{x^i}{i!}$$ $$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$$ So my question is: Why does… ...
1
vote
1answer
34 views

How to show $\gamma =\sum_{m=2}^{\infty}(-1)^{m}\frac{\zeta (m)}{m}$?

How to show $$\gamma =\sum_{m=2}^{\infty}(-1)^{m}\frac{\zeta (m)}{m}$$ where $\gamma $ is the Euler-Mascheroni constant and $\zeta (m)$ is the Riemann Zeta Function.
1
vote
0answers
31 views

About Abel Summation

http://arxiv.org/pdf/math/0504289v3.pdf Here i'm trying to understand page 5. Writer uses the abel sum to find the sum of the prime's reciprocals. So he founds the formula (2.2.1) Now here y=2 ...
0
votes
1answer
35 views

How do i prove the Euler-Mascheroni constant is positive?

Define $\gamma=\lim_{n\to\infty} \sum_{k=1}^n 1/k - ln(n)$. I know that $\gamma$ is nonnegative, but i don't know how to prove that it is positive.
0
votes
0answers
43 views

How to show that $-\int_0^\infty{\ln(x)\exp(-x)dx}=\gamma$ where $\gamma$ is the Euler's constant.

I know that the Euler's constant $\gamma$ is defined by the term $\lim\limits_{N\rightarrow\infty}{\left(\sum_{n=1}^{N}{\frac{1}{n}}-\ln N\right)}$. But I can't see how this term is equal to the ...
0
votes
2answers
38 views

How to plot a curve in complex plane that include e constant

I can understand how a complex numbers such as $10 + 7i$ can be plotted in complex plane with Imaginary and Real axis. But I have no idea how to approach this problem. I have been asked to plot ...
5
votes
6answers
4k views

Is the integral of $\frac{1}{x}$ equal to $\ln(x)$ or $\ln(|x|)$?

The inconsistency I see between mathematical subjects is really confusing me. I understand that it isn't possible for $e^x$ to be less than zero for real $x$, which is probably why they say that the ...
1
vote
2answers
113 views

Euler-Mascheroni constant: understanding why $\lim_{m\rightarrow \infty} \sum_{n=1}^{m} (\ln (1 + \frac{1}{n})-\frac{1}{n+1})= 1 - \gamma$

I am trying to understand why the Euler-Mascheroni constant $\displaystyle \gamma = \lim_{n \rightarrow \infty} \left ( \sum_{k=1}^n \frac{1}{k} - \ln n \right )$ is equal to $1 - \displaystyle ...
2
votes
2answers
65 views

Evaluate $a^i$ where $a$ is real:

I have a question involving the evaluation of $3^i$, but I am unsure how to do this. I know how to solve such questions involving $e^{i\theta}$, but how does this work with a different base? (I ...
2
votes
1answer
42 views

An Euler-Mascheroni-like sequence [duplicate]

How does one compute the limit of the sequence: $$\sum_{k = 0}^{n}\frac{1}{3k+1} - \frac{\ln(n)}{3}$$ I would apreciate a hint.
1
vote
1answer
37 views

Identify Equation

I have a mathematical equation. Can anyone tell the function related to the equation? $$ \frac{1}{1+\exp\frac{x-u}{s}}$$ Any help is much appreciated. Thanks.
0
votes
2answers
67 views

Asymtptotic limit of $e^x$ [closed]

I am looking for functions $A,B$ such that $$ A < e^x < B.$$ $A,B$ should be as close to $e^x$ as possible. I was trying to find something, but all I found was very distant. Can someone suggest ...
1
vote
1answer
47 views

Outputting inequality with $e^x$

I many books I can find inequality which estimates $e$: $$\left(1+\frac{1}{n}\right)^n \lt e \lt \left(1+\frac{1}{n}\right)^{n+1}$$ I am wondering if correct is also to write: ...
13
votes
1answer
261 views

Prove that $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor=\gamma$

Prove that $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor=\gamma$$ Can we find a known value for ...
0
votes
1answer
117 views

What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series (An = 1 + 1/2 + ... + 1/n) and this natural logarithm L = ...
0
votes
1answer
49 views

Showing the correctness of inequality with $e$

Is true that always for every $x$ and $n, n\not=0$ this inequality is true: $$\left(1-\frac{1}{n}\right)^{x} \leq e^{-x/n}\;\;?$$ I have the doubt about the values of $n$ which are very close to $0$, ...
2
votes
3answers
84 views

More accurate estimation of mathematical constant $e$

Very often in books and also on Wikipedia we can find that: $$e \approx \left(1+\frac{1}{n}\right)^n$$ but I want more accurate estimation, it means instead using $\approx$ I wonder if I can use ...
0
votes
1answer
50 views

Arp Equation, simple, b = 0

I am working with Arp Equations. Letters slightly changed for simplicity/formatting purposes. \begin{equation} q(t) = \frac{A}{(1+bDt)^{1/b}} \end{equation} now when $b = 0$, how does the result end ...
40
votes
10answers
2k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
1
vote
1answer
79 views

Practical significance of $e$ [duplicate]

We know, for example, the constant $\pi$ is the perimeter of a circle with diameter $1$ unit. In the similar manner how would we explain the constant $e$. I have searched a lot for it. But I couldn't ...
0
votes
2answers
55 views

Simplifying euler exponent?

How would I go about simplifying and finding the exact value for this question: $e^{6\ln(4)}$ I know that $e ^{\ln x} = x$ but how does the $6$ affect this answer?
8
votes
2answers
110 views

Why is $-\gamma = \int_0^1 \frac{e^{-z}-1}{z}dz+\int_1^\infty \frac{e^{-z}}{z}dz$

It seems like the sum of the two RHS integrals is "well known"$^\dagger$ to be Euler's constant: $$\gamma \equiv \int_1^\infty \frac{1}{\lfloor z\rfloor} - \frac{1}{z}dz \quad\stackrel{?}{=}\quad ...
1
vote
1answer
89 views

Proving this identity $\gamma=1+\ln(\frac{1}{2})+\sum_{k=1}^{\infty}(-1)^{k+1}\frac{\zeta(k+1)-1}{k+1}$ where $\gamma$ is the Euler-Masceroni constant

I've seen this identity here $$ \displaystyle \gamma=1+\ln(\frac{1}{2})+\sum_{k=1}^{\infty}(-1)^{k+1}\dfrac{\zeta(k+1)-1}{k+1} $$ and I'd like to know how it is deduced. Could anyone help? Thanks. ...
0
votes
0answers
88 views

Continued fraction of $\gamma+1$ using recursion

Number $\gamma,$ the Euler-Mascheroni constant, is defined as the value of $$\gamma = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k} - \ln(n).$$ We know that $$\lim_{n\to\infty} ...
1
vote
4answers
565 views

Explain this chain rule for differentiating $y=xe^{-kx}$

I am asked to differentiate $$y=xe^{-kx}$$ The answer I am given is $$e^{-kx}(-kx+1)$$ I understand that when e is differentiated, it remains the same. I also see the product rule, but I'm not sure ...