Jack D'Aurizio
Reputation
100/100 score
 9h answered Calculating the value of $\cos \left(\frac{1}{2} \arccos \frac{3}{5}\right)$ 10h comment Primitive elements proof possible duplicate of Proving that the multiplicative group mod p (p is prime) is cyclic 11h answered $\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$ 12h comment What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$? In fact, the behaviour of $\frac{1^n+2^n+\ldots+n^n}{n^{n+1}}$ is not $\approx\frac{1}{n+1}$ but $\approx\frac{e}{n(e-1)}$. 12h comment What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$? @Omnomnomnom: however, Soke is right in glancing that there is something quite subtle, since $n$ is both the number of intervals we are considering and the exponent of the integrand function, and they are tending to infinity together. 12h comment Contest Problem Oh, pardon, my bad, I misread the question. I thought we were dealing with all the positive integer numbers starting and ending with a $4$. 12h answered Prove this inequality $25ab+25a+10b\le38$ 12h comment What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$? @StevenStadnicki: well, in that case one may prove that $$f(m,n) = \frac{1^m+\ldots+n^m}{n^{m+1}}$$ is monotonic with respect to both $m$ and $n$ and exploit this fact to be allowed to exchange limits. 12h comment Contest Problem How do you pick a random element from an infinite set? 12h comment What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$? @RazvanParaschiv: my last line does not involve any limit, it is just an inequality holding for every $n\geq 1$. The trick is to consider limits just after that and state that the limit is zero by squeezing, since $$\lim_{n\to +\infty}\frac{e}{n(e-1)}=0,$$ no issues. 12h answered This one wierd trick integrates fractals. But does it deliver the correct results? 12h comment What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$? I like it too. I think I have managed to find another proof, similar to yours, and without integrals at all. Just see below. 12h revised What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$? added 233 characters in body 13h comment $\sum_{i=1}^n\frac{1}{y_i^2}=1$ and $y_{min} \rightarrow \infty$, prove $\lim_{n \rightarrow \infty }\sum_{i=1}^n e^{-y_i}=0$ Exactly. We have to know how fast the minimum of the $y_i$s goes to infinity depending on $n$. Otherwise, we can choose a minimum, then the number of variables (a huge number), and the last limit is not zero. 13h comment What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$? (+1) This is probably the fastest way. 13h answered What is the limit $\lim_{n\to\infty}\frac{1^n+2^n+3^n+\dots+n^n}{n^{n+1}}$? 13h comment There is an estimation of a $\sum_{x\in\mathbb{F}_p} \left(\frac{a_{n}x^n+a_{n-1}x^{n-1}+…a_0}{p} \right) \le (n-1)\sqrt{p}$ terrytao.wordpress.com/tag/hasse-weil-bound 13h comment There is an estimation of a $\sum_{x\in\mathbb{F}_p} \left(\frac{a_{n}x^n+a_{n-1}x^{n-1}+…a_0}{p} \right) \le (n-1)\sqrt{p}$ en.wikipedia.org/wiki/Weyl%27s_inequality 13h comment Evaluate $\lim_{x\rightarrow \infty}(1+\frac{1}{\sqrt{x}})^{\sqrt{x}}$. Euler's Limit $\sqrt{x}$ does not exist when $x<0$ and $+\infty$ can be approached only from the left. 1d awarded logarithms