The formula for the  the sum of infinite terms of a geometric progression  is $S_{\infty} = \dfrac{a}{1 - r}$, where $r &lt; 1$. Why is this only defined for $r &lt; 1$? My teacher doesn't know why.
When is the formula $$S_{\infty} = \dfrac{a}{1-r}$$ valid?   Does |$r| &lt;1$?
How to get only one vertically centered equation number in align environment with two equations There is a type and search drodown which will appear after user type in the value. I need to click on the dropdown which contains the name of the drop down field.   The problem here is that there is no unique attribute present (No unique class, no unique name) &amp; the id keeps on changing after every login so cannot take id also. The choice of measurement basis on one half of an entangled state affects the other half. Can this be used to communicate faster than light? It is often stated, particularly in popular physics articles and videos, that if one measures a particle A that is entangled with some other particle B, then this measurement will immediately affect the state of the entangled partner.  For example, if Alice and Bob share an entangled pair of electrons and Alice measures her spin in the $x$ direction, then Bob's spin will also end up spinning in that direction, and similarly if she measures in the $z$ direction. Moreover, the effect will be instantaneous, regardless of the spatial distance between the two particles, which seems at odds with special relativity.  Can I use a scheme like this to communicate faster than light? Start script on open blender file Recover data from SD card Is it possible to configure Windows 7 to use YYYY-MM-DD instead of DD/MM/YYYY as the date format? How? Where? Is a DNA molecule a single strand of polynucleotide or two of them linked together? Remove a part of a list of names I have some files and each files have distinct name but all of them have the same "-statement" at the end like this:  flower-statement.mp3 leave-statement.mp3  ball-statement.mp3 goal-statement.mp3   So I want to remove this: -statement  Is that possible? So, if we have a large pipe which is constricted in the middle, the fluid in the middle section  will move faster. Why is it so? What dictates that liquid that an X amount of it must pass on the other side regardless of the size of the encasement?  The only reason I could think about is because that the initial pressure which causes the movement somehow is being transferred to a smaller area, but that makes no sense. I'm trying to show that $R = \mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD, but I got stuck.  To prove this, I could try to find two "different" factorisations for one element, but I am not familiar with this, so I tried to use a lemma and one of the previous exercises. If my syllabus is right, in every UFD counts $$ x \ \text{is irreducible} \quad \iff \quad x \ \text{is prime}$$ My syllabus also states that the elements $\bar{X}, \bar{Y} \in R$ are irreducible. So I tried to show that at least one of the elements $\bar{X}, \bar{Y} \in R$ does not generate a prime ideal.  This would mean that I should find two polynomials $f,g \in \mathbb{Q}[X,Y]$, such that  $$\exists p \in \mathbb{Q}[X,Y], \quad fg - pX \in (Y^2-X^3)$$ and $$\forall q \in \mathbb{Q}[X,Y], \quad f-qX \notin (Y^2-X^3) \ \wedge \ g-qX \notin(Y^2-X^3)$$ or the same thing but then for $Y$.  I hope that you can tell me if this approach is correct, and provide me a hint. I'd appreciate it if you told me the solution, but please start your answer with a hint, clearly separated from the rest. Derivation of Newton-Euler equations I am in search of a simplified version of the derivation of  (both translational and rotational) for a rigid body (3D block) that has a body fixed frame and where the center of mass of the body is not at the center of gravity.  I can find elementary derivations for the same system when the center of mass is at the center of gravity, but not for my system in question.    I am using the derivation as background research for a rotordynamics project that I'm working on.  Any help and or references would be greatly appreciated!  :) Visiting Russia as an Arab. Am I going to be hassled? I'd like to setup a WCF service to send emails. The System.Net.MailMessage doesn't seem to be serializable, and cannot be passed in a [DataContract]  The error I receive is      Type 'System.Net.Mail.MailAddress'   cannot be serialized. Consider marking   it with the DataContractAttribute   Any suggestions? Bending of light - photon's inertia instead of mass This question is NOT about access to USB or external drives.  WPS Witer and other just installed Ubuntu software can NOT open files saved on my /media directory. I keep getting the annoying message: « Could not read the contents of '/media': Permission denied ».   MY SIMPLE QUESTION REMAINS:  How can I access all files and folders on my user username/media/ directory?  I'm using Ubuntu 18.04.3 LTS.  Ubuntu's Nautilus can indeed access all files and folders in all locations. However, Ubuntu's software that I have downloaded (e.g. KRITA, WPS WRITER etc) can NOT access any file or folder on the /media directory. Thanks in advance. How can you get the names of method parameters? If I have a method such as:  public void MyMethod(int arg1, string arg2)   How would I go about getting the actual names of the arguments? I can't seem to find anything in the MethodInfo which will actually give me the name of the parameter.  I would like to write a method which looks like this:  public static string GetParamName(MethodInfo method, int index)   So if I called this method with:  string name = GetParamName(MyMethod, 0)   it would return "arg1". Is this possible? Azimuth and Distance, NumericalDigitize, Numerical Vertex Edit, XY tools and other plugins are not on QGIS Plugin Manager version 2.0.1 Will these plugins no longer be available in QGIS 2.0.1? Random keys on keyboard stopped working Understanding NSString comparison How would I solve the following problem involving elementary matrices? Let $A=\begin{pmatrix}0&amp;5\\ 7&amp;4\end{pmatrix}$.      1) Write $A$ as a product of $4$ elementary matrices.      2) Write $A^{-1}$ as a product of $4$ elementary matrices.   My work. I have managed to find $A^{-1}$, which came out to be this:  \begin{pmatrix}-\frac{4}{35}&amp;\frac{1}{7}\\ \frac{1}{5}&amp;0\end{pmatrix}  However, I am struggling to figure out how I would split each of these matrices into $4$ elementary matrices. Any help?