Questions relating to Euler's Constant (commonly written as $\gamma$), the constant defined as the difference of the natural logarithm and the harmonic numbers as they get (asymptotically) larger. $$\gamma=\lim_{n \to \infty}H_n-\log n$$ where $$H_n=1+\frac{1}{2}+\cdots+\frac{1}{n}$$
33
votes
10answers
1k views
What it the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?
What it the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$?
Using the definition:
$$\lim_{n\rightarrow\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$
I finally get 2 ...
10
votes
8answers
600 views
“How I wish I could calculate pi” analogs…
You might know the mnemonic for $\pi$ in the title or even this more elaborated one:
Sir, I bear a rhyme excelling
In mystic force, and magic spelling
Celestial sprites elucidate
All my own ...
7
votes
2answers
548 views
Has Euler's Constant $\gamma$ been proven to be irrational?
I found a paper by Kaida Shi called "A Proof: Euler’s Constant γ is an Irrational Number" which claims to have proven the irrationality of $\gamma$.
I know people have been trying to prove that ...
5
votes
3answers
279 views
Limit of Zeta function
I'm looking for a reference for (or an elementary proof of)
$$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$
Thanks for your help.
5
votes
2answers
307 views
Equality with Euler–Mascheroni constant
While trying to prove integral with exponential function and logarithm in an alternative way, I came to this solution:
$$\sum_{k=0}^{+\infty}(-1)^{k+1}\frac{\log (k+1)+\gamma }{(k+1)}.$$
As both ...
5
votes
1answer
120 views
Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind
Consider the integral
$$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x$$
For $n=3$ we have
...
4
votes
6answers
831 views
Is the integral of 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 ...
4
votes
4answers
191 views
Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?
Of all the possible combinations of positive numbers that sum to 10, which has the largest multiplication?
I had also got a clue: it's related to e.
Please help! ...
4
votes
1answer
131 views
Euler's Question
I came across this problem that I would like to ask you about:
For which values $a>0$ does there $\exists$ a limit of the sequence $$a, a^{a},a^{a^{a}}, a^{a^{a^{a}}}...$$
Well this looks like a ...
3
votes
2answers
138 views
Potence of Euler's Number
Show with help of the Bernoulli Inequality that
$$\lim_{n\rightarrow\infty}\left(1-\frac{1}{n^2}\right)^{n}=1$$
End with:
$$\lim_{n\rightarrow\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}$$
3
votes
2answers
98 views
Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant
Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution )
...
2
votes
3answers
85 views
Limit of an equation similar to the Euler's constant definition
$$ \lim_{n\to \infty} \left(2+\frac{1}{n}\right)^{n} = ? $$
I don't know even how to start this.
2
votes
1answer
56 views
Does this have a name? (Regarding ways to calculate e)
Just wondering...came across this relationship regarding Euler's number in my math tinkerings, but I'm unaware if this particular relationship has a specific name or not:
...
2
votes
0answers
89 views
Integral representation of Euler's constant
Prove that : $$ \gamma=-\int_0^{1}\ln \ln \left ( \frac{1}{x} \right) \ \mathrm{d}x.$$
where $\gamma$ is Euler's constant ($\gamma \approx 0.57721$).
This integral was mentioned in Wikipedia as in ...
2
votes
1answer
39 views
Terms that cannot be solved for a variable
Yesterday our analysis professor told us you cannot solve
$$
y = e^x+2/(1+x^2)
$$
for x, but you have the option to approximate this numerically. He did not prove that, he just noted it.
I can't ...
1
vote
1answer
63 views
exponential population growth models using $e$?
Im trying to understand this write up [1] of cell population growth models and am confused about the use of natural logarithms. If cells double at a constant rate starting from 1 cell, then their cell ...
1
vote
1answer
62 views
Use Euler's method with step size 10^-n to estimate x(1), where f(x) is the solution of the initial-value problem below. f(x)=-x x(0)=1
Use Euler's method with step size $10^{-n}$ for $n=1,2,3,4.$ to estimate $x(1)$, where $f(x)$ is the solution of the initial-value problem below.
$x'=f(x)=-x$
$x(0)=1$
EDIT / UPDATE:
x_n+1=x_n + ...
0
votes
0answers
95 views
Help with interesting sum involving Euler's constant
Wikipedia gives an interesting infinite sum for Euler's constant $\gamma$ and I was wondering how one would evaluate this interesting sum. The sum is given as follows:
Let $N_0 (x)$ and $N_1 (x)$ ...
