Why this integral diverges intuitively speaking I'm reading Stewart's calculus book and I faced with an intuitive issue.  Intuitively, I don't understand why since both graphs looks like the same up to a zoom in/out. Could someone explain to me this intuitively? For me both should be convergent in this context.  It's difficult to grasp this intuitively because it's like both functions are approaching to zero without the area of the second one approaching to zero, how is that possible?
Geometric intuition of improper integrals I am aware that the area under the curve of $\frac{1}{x}$ is infinite yet the area under the curve of $\frac{1}{x^2}$ is finite.   Calculus and series wise, I understand what is going on, but I can't seem to get a good geometric intuition of the problem. Both curves can be shown to converge to $0$ (the curves themselves, not the area), and on the interval from $1$ to infinity, the two curves have nothing intrinsically different. Can someone please provide me with an good geometric intuition of what's going on? I can't find anything on the web, people seem to not want to explain it geometrically.
What is spin as it relates to subatomic particles? What is the quantum mechanical explanation of the octet rule? What is the quantum mechanical explanation of the ? In other words, what makes the octet rule be true from a quantum mechanical view? How we explain  what makes some atoms don't follow the octet rule using QM? I'm currently taking Quantum Mechanics in school and I came across a table displaying the following in the given order:   $E_n (\text{energy levels}), g_n (\text{eigen functions})$   $n \ l, [m]$  $E_1, 1$   $1s, [0]$  $E_2, 4$   $2s, [0]$ $2p, [-1,0,1]$  $E_3, 9$   $3s, [0]$ $3p, [-1,0,1]$ $3d, [-2,-1,0,1,2]$    Now, as I understand $n$ defines the size of the orbital, $l$ the shape and $m$ the orientation.  My question is, how does the atoms shell model fit into this if one can have two electrons per orbital? Are $1s$ the same as the orbital $2s$, but just renamed for the sake of higher energy levels? Or do two similarly looking orbits exist for different energy levels? For example two "layers" of spherical electron clouds for an atom with $4$ electrons.  I just can't see how $2$, $8$, $18$ becomes $2$, $8$, $8$ as we were taught in Physics at high school? Am I missing something? When I upload an image to Stack Overflow, I cannot see my JPEG file in my post. It's all blank. I've tried it with different files, but gotten the same result. How to get instance variables in Python? Playing a remote movie on the remote computer How can I make a fake name when I use command blocks? I'm trying to make a mini game but I want it so that when you press a button over a command block it will say "BOB welcome", so 'bob' instead of the @a or @p sign and then followed by the message. All I know is that I need to use /tellraw. X,Y,Z missing in F3 When I press F3 to see the debug screen in I use Minecraft 1.8.1, some information is missing.   Most notably, there are no X, Y, Z coordinates.     What happened to my debug overlay? If $k=1$, we have \begin{align} \sum_{n\ge1}\frac{1}{\binom{2n}{n}}&amp;=\frac12\sum_{n\ge1}n\mathrm{B}(n,n)\\ &amp;=\frac12\int _{0}^{1}\sum_{n\ge1}n(t-t^2)^{{n-1}}{\mathrm  d}t\\ &amp;=\frac12\int _{0}^{1}\frac{1}{(t-t^2-1)^2}{\mathrm  d}t\\ &amp;=\frac13+\frac{2\sqrt3\pi}{27} \end{align} I wonder if there exists a general formula of $\sum_{n\ge1}{\binom{2n}{n}^{-k}}$ with $k\in\Bbb N^*$. This may involve some ${}_pF_q$ functions. Ever since mods gained the ability to undelete comments, deleting them has become easier as the confirmation dialog was removed.  However, this is not the case in the mod queue &mdash; I still get a confirmation dialog for each comment I want to delete in the queue under a custom-flagged post (single-comment flags can be deleted just fine). And since this is the place from where I delete comments the most, it can be annoying.  In addition, the "purge comments" button, when invoked from the queue, refreshes it, so I have to scroll back down, remember which post I just purged, and mark the flag as helpful. Usually not a big issue but if there are multiple "obsolete comments" flags, then it can get confusing.  Could the workflow for comment thread moderation from within the queue be improved? Ideally, make comment deletion work like everywhere else on the site, and make the purge button AJAX-based like the comment deletes. Is the claim that functions of independent random variables are themselves independent, true?   I have seen that result often used implicitly in some proofs, for example in the proof of independence between the sample mean and the sample variance of a normal distribution, but I have not been able to find justification for it. It seems that some authors take it as given but I am not certain that this is always the case. What does the big intersection or union sign of a set mean? In some cases, we will have to consider the union or the intersection of several, even infinitely many sets, defined in the obvious way. For example, if for every positive integer $n$, we are given a set $S_n$, then   $$\bbox[border:1px solid red]{\bigcup_{n=1}^\infty S_n}=S_1\cup S_2\cup\cdots=\{x\mid x\in S_n\text{ for some }n\},$$   and   $$\bigcap_{n=1}^\infty S_n=S_1\cap S_2\cap\cdots=\{x\mid x\in S_n\text{ for all }n\}.$$   Two sets are said to be disjoint if their intersection is empty. More generally, several sets are said to be disjoint if no two of them have a common element. A collection of sets is said to be a partition of a set $S$ if the sets in the collection are disjoint and their union is $S$.   Normally what I know is that you can make a union or an intersection between only two sets. In this expression, there is a big union of sets. I'm asking about the meaning of this expression – what does it mean? What does the infinity sign do at the top?  Things are even more complicated with De Morgan's laws, which use the same expression:     Two particularly useful properties are given by De Morgan's laws which state that   $$\left(\bigcup_nS_n\right)^c=\bigcap_nS_n^c,\quad\quad\quad\quad\left(\bigcap_nS_n\right)^c=\bigcup_nS_n^c.$$   Anyone who can explain to me the expression or De Morgan's laws would be much appreciated. "PuTTY key format too new" when using ppk file for PuTTY SSH key authentication I wanted to connect to my Unix server using SSH keys. For that I had converted my id_rsa file from ./ssh directory to a ppk file using PuTTYgen, but after loading my .ppk key I get this warning:  PuTTY key format too new   If I use my default id_rsa file, then I get error:  OpenSSH SSH-2 private key (old PEM format)  Either way I am getting stuck. Can anyone please help me? Where can questions about our community-run moderation tools be asked? The question  was recently closed due to not being "about the software that powers the Stack Exchange network, within the scope defined in the help center." MEE is Johann Gambolputty made a comment that     SmokeDetector is a community run project to help to fight spam. Therefore your question is unfortunately off-topic for this site. You can find help in  chatroom.   If questions about community run moderation bots (e.g. SmokeDetector, FireAlarm, Natty, Queen, etc.) are off-topic here on Meta Stack Exchange, is there anywhere on the Stack Exchange network where one can post a Question and Answer-style question on them, or are they only on-topic in appropriate chat rooms? Proving $ \binom n 0 ^2 + \binom n 1 ^2 + \dots + \binom n n ^2 = \binom { 2n} n $ without induction I have to prove that: $$ \binom n 0 ^2  + \binom n 1 ^2  + \dots + \binom n n ^2 = \binom { 2n} n $$  I don't want a complete solution, but only a hint. Series $\frac{x^{3n}}{(3n)!} $ find sum using differentiation How to count the node a user has created in a View? I would like to take a picture of a street during the day to depict how much traffic and cars there are on the street.  Is there anyway to do this with a long exposure photograph?  I have a Nikon D90 and I set the f stop down to 1/16 (the lowest of my lens) and set the time to 2 seconds. Doing this, the exposure is almost saturated when taken around 5pm with the sun setting behind a tree. Doing this, the cars passing by were very hard to make out against the stationary background, being visible as a blur in the color of the car.  Perhaps this is not possible and I should just do this at night where the cars lights are much brighter than the background.  Please suggest how else I might depict the traffic in a photograph?  Are daytime long exposure photographs even possible? How to install Flash player on Ubuntu? Even though I have the latest Adobe Flash player 11.2 for Linux in Firefox -   - a particular online flash game still requests that I update to the latest flash player.  I've tried this online game on my laptop (Windows 8 with Flash Player 11.8) and it has no problems.  Is there a way to trick Ubuntu or this game into thinking Flash Player 11.8 is "installed"? check if Gutenberg is currently in use How can I check if the editor that is currently being used is Gutenberg in a WordPress plugin?  I need this because Gutenberg lacks post_submitbox_misc_actions, so I need a fallback which will only be used if the current editor is Gutenberg.