Linked Questions

30
votes
5answers
11k views

How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
18
votes
3answers
1k views

Intuitively, why is the Euler-Mascheroni constant near $\sqrt{1/3}$?

Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting. Some time ago, I was struck by the coincidence that the Euler-Mascheroni ...
3
votes
3answers
442 views

Evaluation of Euler's Constant $\gamma$

Long back I had seen (in some obscure book) a formula to calculate the value of Euler's constant $\gamma$ based on a table of values of Riemann zeta function $\zeta(s)$. I am not able to recall the ...
4
votes
3answers
829 views

Euler-Mascheroni constant expression, further simplification

The Euler-Mascheroni constant gamma is defined as: $$\gamma=\lim\limits_{n \rightarrow \infty}\left(\sum\limits_{m=1}^{n} \frac{1}{m} - \log(n)\right)$$ From this previous question Do these series ...
5
votes
3answers
800 views

$\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$

I should be able to answer this myself, but feel insecure anyway. I want to know, whether a function f(n) is bounded if n goes to infinity (and if it's bounded, the limit). Heuristically it appears (...
2
votes
1answer
263 views

Re-Expressing the Digamma

I was reading some articles on the digamma function, and I was wondering if anyone knows how to express the digamma function $\psi^{(0)}(n)$ in terms of a trigonometric function or a logarithmic ...
1
vote
2answers
333 views

Question on Macys formula for Euler-Mascheroni Constant $\gamma$

I think that: $\gamma = \lim_{n\rightarrow\infty} ~~~ 2H_{n} - H_{n(n+1)}~~~~~~$ (where $H_{n}$ is the $n$-th harmonic number) is a closed form of Macys $\gamma$ formula: $\gamma = \lim_{n\...
2
votes
0answers
181 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} \frac{n}{\pi(n)}-\ln(n)=-...
1
vote
0answers
153 views

Series for Stieltjes constants from $\gamma= \sum_{n=1}^\infty \left(\frac{2}{n}-\sum_{j=n(n-1)+1}^{n(n+1)} \frac{1}{j}\right)$

Euler's constant has the following representations (Euler-Mascheroni constant expression, further simplification, https://math.stackexchange.com/a/129808/134791, Question on Macys formula for Euler-...
0
votes
0answers
79 views

What is the series representation of $\frac{1}{\gamma}$?

If a series representation of ${\gamma}$ The Euler-Mascheroni constant is:$\displaystyle\sum_{k=1}^{+\infty}\left[\frac 1 k -\log\left(1+\frac 1 k\right)\right]$ then what is the series representation ...