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

learn more… | top users | synonyms

1
vote
1answer
34 views

Find the region R for which the sequence converges

Find the region $(x,y) \in R$ for which the following sequence converges $$\left| e^n\frac{(\sqrt{y}-\sqrt{x})^{2n}}{x^n} \right| \to 0$$ I am currently doing number theory research on studying the ...
2
votes
0answers
184 views

Why is $e$ close to $H_8$, closer to $H_8\left(1+\frac{1}{80^2}\right)$ and even closer to $\gamma+log\left(\frac{17}{2}\right) +\frac{1}{10^3}$?

The eighth harmonic number happens to be close to $e$. $$e\approx2.71(8)$$ $$H_8=\sum_{k=1}^8 \frac{1}{k}=\frac{761}{280}\approx2.71(7)$$ This leads to the almost-integer ...
2
votes
0answers
99 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 ...
1
vote
0answers
63 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, http://math.stackexchange.com/a/129808/134791, Question on Macys formula for ...
1
vote
0answers
64 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
0answers
112 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
0answers
172 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 $N_1 (x)$ ...
0
votes
0answers
18 views

How to simplify this representation of $\gamma$?

I have a function of two variables, $Z(n,m) = (-1)^m \left( \frac{1}{m-2^n} + \frac{1}{2^n \log (2^n /m)} \right)$, and the infinite sum $\sum_{n=1}^{\infty} \sum_{m=1}^{2^n -1} Z(n,m) = \gamma$, the ...
0
votes
0answers
37 views

Transform cos to e function

What are the steps in order to transform the cosine function to the exponential function: $$ \left[\cos \left(\frac{k \pi} N\right)\right]^n \approx e ^ {\frac{-n}2 \left(\frac{k \pi} N \right)^2} ...
0
votes
0answers
45 views

Evaluating $\int{e^{-t^{2}}\,dt}$

Let $f(x) = e^{x}$, which, expressed as a Maclaurin series, is equal to: $$\sum_{i = 0}^{\infty}{\frac{x^{i}}{i!}}$$ Therefore, $f(-t^{2})$ gives: $$\sum_{i = 0}^{\infty}{\frac{(-t^{2})^{i}}{i!}} = ...
0
votes
0answers
123 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 ...