@user60887 Use dem definitions. $\lfloor x\rfloor=n$ is the unique integer such that $n\leqslant x<n+1$.
Note that if $x'=x+m$ with $m$ an integer, then $n+m$ has this propertyw.r.t. $x'$, simply by summing in the above. Thus $\lfloor x+m\rfloor =\lfloor x\rfloor +m$
Hi! I have a little question on the $\TeX$ notation on the main site. It isn't possible to mark more than one line by "\tag{n}", what should I do to enumerate more than one line in a formula? What I'm doing now is adding "\quad (n)" after each line to be enumerated
@skullpatrol As regards this question all I know is that some of the students that received this question made no progress. They didn't even know how to tackle it. It's sad but it's true.
@Alizter Your double sum is: $$\sum_{k=0}^\infty \tfrac{(-1)^k}{k!}\sum_{n\geq k} \frac{(-1)^n}{e^n}$$ use the geometric sum there, to get $$\sum_{n\geq k} \frac{(-1)^n}{e^n} = (-1/e)^k / (1+1/e)$$ and then sum the exponential
@Alizter You were summing first over $n$, and then over the $k$ satisfying $k\leq n$, right? .. if you do it the opposite way, sum over $k$ and then for the $n$ greater than that $k$.
So you'd get $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^{n-k}}{e^n k!} = \frac{e^{1/e}}{1+e^{-1}} $$
@PaulRS I am not too familiar with geometric series so can you go through the steps a bit more? On your second step can you explain how that becomes that?
@Alizter You need to check this because the left side is approximately $2.6618257053804178285$ and the right side is approximaely $5.7417551233587916696$
@skullpatrol I also proposed for a high school contest the following question. Compute $$\sum_{n=1}^{\infty}\arctan\left(\frac{1}{(2+\sqrt{3})n^2+(2+\sqrt{3})n+2-\sqrt{3}}\right)$$ and express the answer in terms of $\pi$
@skullpatrol from the feedback I received so far I can say that most probably it will be rejected.
@skullpatrol the teacher I talked to failed to compute it and then I sent him my solution. (or maybe he was too lazy to compute it)
It's said the mathematicians are lazy persons, you know. :-)
@leo It wasn't intended to be a multiplication table; It's supposed to be a power table... but I think I realize now that the top row should have been $1\cdots 6$ instead of the elelements in $\mathbb{Z}_9^*$
@PaulRS I'm aware how to test if a number is a generator... just raise each element in the set $\mathbb{Z}_m^*$ to every power $n$ such that $1\leq n\leq \varphi(m)$ and see if you get every element in the set back... but I was wondering if there's a direct way to find a generator without having to test every element in the group.
@agent154 There is a randomized algorithm I know... which works well if you can factorize $\phi(n)$, it should be pretty fast, but... that is good for a computer, not so much for a human I think
I'm only worried about being able to solve questions on a test quickly... but I guess the questions won't be hard enough such that I need to test that many possiblities
I have a test in 30 minutes and I'm trying to practice how to find generators of cyclic groups
@PaulRS So for $\mathbb{Z}_9^*$, then I only need to check $2$, $5$, $7$, since $1$ is never a generator, and $4$ and $8$ are powers of $2$? Could it ever be a case where $4$ is a generator but not $2$?
@charlie my first advanced calculus (real analysis) test is in less than two weeks! Nov 08is when I get to show whether I'm the real deal or I am only a squeal
I have an equation: $ x+y+z+t = 7$. I want to know how many solutions does this equation have? $x,y,z$, and $t$ are positive integers. I have no idea how to solve this. Can you please help me to solve this question?
Suppose $Y \subset X$. A subset $E$ of $Y$ is open relative to $Y$ iff $E = Y \cap G$ for some open subset $G$ of $X$. Does this definition imply $E = Y$?
I need to determine the point group of the following patterns:
I think the one on the left is $C_1$ and the one on the right is $D_2$
I have an equation: $ x+y+z+t = 7$. I want to know how many solutions does this equation have? $x,y,z$, and $t$ are positive integers. I have no idea how to solve this. Can you please help me to solve this question?
