Is there any formula to find out the summation of the series.   $$\sum_{i=1}^{n} \lfloor \frac{n}{i} \rfloor$$ Can someone help me with this.
$$ \sum_{i=1}^N\left\lfloor\frac{N}{i}\right\rfloor $$ Is there a closed form expression to the above sum? (Mathematica doesn't give me anything)
Integral of $\frac{1}{x^4+1}$ Just doing this for revision, seems much harder than it should be, should I use $x=\tan u$ ? Any help appreciated. Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof? "They are quite full up this year." Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$. Problem: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  My work: So I think I have to do a proof by induction and I just wanted some help editing my proof.  My attempt:  Let $P(n)=1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.  Then $$P(1)=1^2=\frac{1(1+1)(2+1)}{6}$$ $$1=\frac{6}{6}.$$ So $P(1)$ is true.  Next suppose that $P(k)=1^2+2^2+\cdots+k^2=\frac{k(k+1)(2k+1)}{6}$ for $k \in \mathbb{N}$. Then adding $(k+1)^2$ to both sides of $P(k)$ we obtain the following: $$1^2+2^2+\cdots+k^2+(k+1)^2=\frac{k(k+1)(2k+1)}{6}+(k+1)^2$$ $$=\frac{2k^3+3k^2+k+6(k^2+2k+1)}{6}$$ $$=\frac{2k^3+9k^2+13k+6}{6}$$ $$=\frac{(k^2+3k+2)(2k+3)}{6}$$ $$=\frac{(k+1)(k+2)(2k+3)}{6}$$ $$=\frac{(k+1)((k+1)+1)(2(k+1)+1)}{6}$$ $$=P(k+1).$$ Thus $P(k)$ is true for $k \in \mathbb{N}$. Hence by mathematical induction, $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ is true for $n \in \mathbb{N}$. Let $(B_t)_{t\geq 0}$ be a standard Brownian motion in $\mathbb{R}^d$. It is intuitive that, for fixed $s&lt;t&lt;u$  $$\mathbb{E}[B_t\mid \sigma(B_s,B_u)]=B_s+\frac{t-s}{u-s}(B_u-B_s).$$  However, I cannot think of a way to show this rigorously.   If first attempted to take $A\in\sigma(B_s,B_u)$ and show that $\mathbb{E}[1_A B_t]=\mathbb{E}[1_A(B_s+\frac{t-s}{u-s}(B_u-B_s))]$. But I cannot manage to show this equality.   I'd be very thankful for any ideas and suggestions on how to tackle this problem. If a function $f$ is continuous $\implies$ $|f|$ is continuous.(Answered by Myself) $(i)$ Given a function $f:E \rightarrow \mathbb{R}$ define $|f|:E \rightarrow \mathbb{R}$ by $|f|(x)=|f(x)|$ for $x \in E$ show if $f$ is continuous at $c \in E$ then so is $|f|$.        $(ii)$ Now the converse is not true, find $f$ such that $|f|$ is continuous but $f$ is not. What is an adjective referring to “someone who puts people down”? I was just thinking about someone I went to school with who had a penchant for putting people down, and I was trying to think of a word that describes this, but couldn’t find it. It would be a synonym of caustic, abrasive, discouraging, and undermining, but these all have slightly different connotations (to wear away", cause discouragement or to make fall). I want an adjective that has the connotation of a hammer hitting you on the head. The word does not necessarily need to describe someone who does this on purpose, out of evil intent. Is there such a word?    The word should give you a mental picture of someone barraging you with negative comments, but it not necessarily affecting you. Think of someone who so often has something negative to “contribute” that many can just either ignore it or roll their eyes at it, and that is the extent of its effect. I keep getting the mental picture of the word buffet as in, how waves buffet the shore. A constant hitting, without the hitting necessarily being devastating in its destruction. What is CTAN.org now? Get rows with most recent date for each different item Principal $n$th root of a complex number This is really two questions.   Is there a definition of the principal $n$th root of a complex number? I can't find it anywhere. Presumably, the usual definition is $[r\exp(i\theta)]^{1/n} = r^{1/n}\exp(i\theta/n)$ for $\theta \in [0,2\pi)$, but I have yet to see this anywhere. Is this because it has bad properties? For instance, according to this definition is it true that for all complex $z$ it holds that $(z^{1/a})^b = (z^b)^{1/a}$? Allow all reviewers to see deleted posts they reviewed So moderators just got . Often, these bans are due to specific very bad reviews, like approving spam or non-answers. It will be common for us to give links to these bad reviews in the ban message that the reviewers will be presented with.  However, many of these reviewers lack the necessary reputation to see deleted posts. If we're banning someone from review for approving spam or non-answers, almost all of the time those posts will be deleted. If we give them links to the reviews they performed, they won't see the now-deleted posts that they approved, and won't have a chance to learn what they did wrong.  As examples of this, I had to actually take a screenshot to show the reviewers  and  the posts that triggered their ban, because they could no longer see them.  I propose that reviewers of any reputation level be able to see posts that they reviewed, even ones that were later deleted, if they are presented with a link to their review. They already saw the post once, so there's no harm in showing it to them again, and reviewers are going to need to see why they made a bad call on a particular review. What is the reason behind "non-static method cannot be referenced from a static context"? Display only the penultimate (second last) row of a text Why can array constants only be used in initializers? I have a big pot full of chicken necks and backs on the stove which I need to separate into (a) stock and (b) everything else.  I'm wondering the best way to go about doing this. I've previously laid cheesecloth into a fine-mesh strainer and poured it through that, but the cheesecloth always seems to kind of clog up, making straining 4 litres of stock a very lengthy process. I also tried a coffee filter laid inside a strainer and got more-or-less the same results.   Alternatively, I ditched the cheesecloth and just used the fine-mesh strainer by itself, but, unsurprisingly, this method leaves some bits I'd rather not have.   I'm wondering: is there a better way? Or should I stick with the cheesecloth and work on being more patient? Is there a workaround to make CSS classes with names that start with numbers valid? "Can not setup lock file:/tmp/g2ipmsg.lock errno : -11 (Resource temporarily unavailable)" I installed IP Messenger (GNOME 2) from Ubuntu software center but its not working. It's always showing an error message on Ubuntu 12.04.  FileName:ipmsg.c Function:create_lock_file Line:157 Can not setup lock file:/tmp/g2ipmsg.lock errno : -11 (Resource temporarily unavailable) What is the term for a factorial type operation, but with summation instead of products? Where does gravitational waves' energy go? A good reference on optimal control theory Ok, so I am reading about decision making and I came across this subject. Fortunately it has a  , but the point is I want to see some examples, and learn to solve regular problems of this field. Of course I can go for the book by Pontryagin but then I don't think that would be an educative book. Does anyone has a suggestion? I would be happy if it also covers LQ problems. P.S. I studied math, therefore I appreciate a book with rigorous approach to the subject.