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
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1answer
150 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 ...
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
1answer
136 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 + ...
3
votes
2answers
161 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: $$\lim_{n\rightarrow\infty}\left(1-\frac{1}{n}\right)^n=\frac{1}{e}$$
4
votes
4answers
395 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! ...
3
votes
2answers
223 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 ) ...
0
votes
0answers
151 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 $N_1 (x)$ ...
5
votes
1answer
146 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 ...
8
votes
2answers
1k 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 ...
39
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 ...
5
votes
2answers
428 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 ...
6
votes
3answers
443 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.