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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1answer
49 views

How to prove this equality about e

I have the assumption that the following holds: $$\lim_{n \to \infty}\frac{1}{n^2} \cdot \sum_{i = 0}^n \left(1 - \frac 1n \right)^i = 1 - \frac{2}{e}.$$ However, I am totally not sure about it. How ...
0
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0answers
40 views

sth interesting about $\gamma$ the euler's Constant

$\gamma $ is one of interesting constants in mathematics, like $\pi$ or $e$ ... But even more strange and mysterious!We even don't know about $\gamma$ whether it is rational or irrational, algebraic ...
0
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0answers
67 views

Consequences of irrationality of e

We know that $e$, $\pi$ are irrational... But WHY do we know it? I am going to give a lecture about irrationality of e, and I'm looking for a reason why it is an interesting subject. I know the proof, ...
5
votes
6answers
202 views

Finding $\sum_{k=1}^{\infty} \left[\frac{1}{2k}-\log \left(1+\frac{1}{2k}\right)\right]$

How do we find $$S=\sum_{k=1}^{\infty} \left[\frac{1}{2k} -\log\left(1+\dfrac{1}{2k}\right)\right]$$ I know that $\displaystyle\sum_{k=1}^{\infty} \left[\frac{1}{k} ...
9
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3answers
145 views

Possible new definition of Gamma (Euler-Mascheroni Constant): $\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$

I think I've discovered a new definition for the Euler-Mascheroni Constant (Gamma) I can't find it online anywhere, has anyone seen it before? $$\lim_{x \to 0} (-\ln ( \sqrt[x]{x!} )) = \gamma$$
2
votes
1answer
65 views

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)…$

Find a closed form for the value of the infinite product $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$ I know it converges. At first I thought it was the Euler–Mascheroni constant, but it's only accurate to about ...
0
votes
1answer
30 views

How to find the inverse function of Euler's number?

Given: $f(x)= \dfrac{e^x}{1+9e^x}$ , what steps would I take to find its inverse? I tried following the steps on finding the inverse of a normal function but I keep getting one of the variables to ...
0
votes
1answer
23 views

Specifying variables in Maple

Two questions 1)How do you express Euler's number (≈2.718) in Maple. For example I want ln(e) to simplify to 1. 2)How do you have a constant with a subscript? For example if I wanted to input the ...
23
votes
6answers
3k views

What Gauss *could* have meant?

I was reading the Wikipedia entry on Euler's identity ($e^{i\pi}+1=0$) and I came across this statement: "The mathematician Carl Friedrich Gauss was reported to have commented that if this formula ...
11
votes
3answers
480 views

Very challenging: max{floor,ceil}=?

I spotted a pattern while trying to generalize a problem. (EDIT: said problem has been removed from this post to avoid confusion. EDIT(2): Here is the problem again: ...
4
votes
1answer
80 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 ...
3
votes
1answer
81 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
142 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
90 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 ...
39
votes
2answers
776 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 ...
5
votes
1answer
189 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} = ...
0
votes
0answers
42 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
68 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
57 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
37 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
172 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 ...
4
votes
2answers
68 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 ...
3
votes
4answers
183 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 ...
0
votes
1answer
41 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
51 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
75 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 ...
0
votes
2answers
53 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?
2
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0answers
34 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 ...
12
votes
1answer
656 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 ...
2
votes
1answer
73 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
54 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
105 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.
0
votes
1answer
49 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
55 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
44 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 ...
1
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0answers
34 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 ...
1
vote
2answers
122 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
67 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.
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: ...
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 ...
0
votes
1answer
148 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
85 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
51 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 ...
2
votes
1answer
938 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 ...