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

0
votes
2answers
63 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 ...
2
votes
1answer
44 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
46 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: ...
12
votes
0answers
137 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 ...
1
vote
0answers
30 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
28 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 ...
0
votes
0answers
32 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
0answers
43 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
0answers
88 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} ...
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 $N_1 (x)$ ...