The value of the series $\sum_{n=1}^{\infty} n/2^{n}$ What is the value of the series  $$\sum_{n=1}^{\infty} \frac{n}{2^{n}}?$$  It is neither geometric nor a telescopic series. How do we approach such kind of series, given that we know that they converge?
How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
$gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$ Let $G$ be a group and $g,h \in G$. I need to prove that if $g$ and $h$ commute and their orders are coprime, then $|gh| = |g||h|$, that is, the order of their product is the multiple of their orders.  Since $gh = hg$, then $(gh)^{ lcm(|g|, |h|)} = e$, so, $|gh|$ divides $lcm(|g|, |h|)$. Since $\gcd(|g|, |h|) = 1$, then $lcm(|g|, |h|) = |g||h|$.  So, $|gh|$ divides $|g||h|$. How do I check if a given string is a legal/valid file name under Windows? I want to include a batch file rename functionality in my application. A user can type a destination filename pattern and (after replacing some wildcards in the pattern) I need to check if it's going to be a legal filename under Windows. I've tried to use regular expression like [a-zA-Z0-9_]+ but it doesn't include many national-specific characters from various languages (e.g. umlauts and so on). What is the best way to do such a check? I have opened a new blender file and deleted the default cube, then imported an obj file . I did not see any thing so i selected it and i zoomed ,nothing appears.In the outliner i saw the object name ,But in the info panel i see that there isn't any verts/faces/edges.  When i opened the obj file in a text editor i found Mesh data in it. I tired this in all blender versions (2.78c/2.78.5/7.9). Can any one help how to view the mesh? EncFs insecure, what to use now I love encfs as it provides file-based encrypting which is quite useful when it comes to cloud storage. But it looks like that especially for this use case, . I'm aware that  but how to deal with it in the meantime? Are there any alternatives that integrate well in ubuntu?  Edit: The security issue I mostly refer to is . It is still present in version 1.8 and makes your files vulnerable if someone get's multiple versions of your encrypted files. If one is worried about services like Dropbox are not that thrustworthy and encrypts folders uploaded into the cloud because of that, the opportunity that the attacker (the service) gets more than one copy of the cyphertext is absolutely given. Minecraft Vanilla 1.9 crash with problematic frame ig8icd64.dll How can I list all user names and/or home directories? I want to list all the users dirs on the machine.Usually, I will do:  ls -l /home   But I use it in a script that will be deployed on others machines and maybe on that machines they don't call it home (e.g. myHome). So I want to generalize it to ls -l ~. But it just lists my users home dirs instead of all users home dirs name (basically I want to get a list of the users names on the machine). How can I generalize it? How to sort an array of integers correctly SMTP 25 closed even though I allow everything in firewall (Debian server running sendmail) I am running sendmail + dovecot in my Debian server. I have enabled the firewall rules to allow everything:   iptables -P INPUT ACCEPT   iptables -P OUTPUT ACCEPT   iptables -P FORWARD ACCEPT   iptables -F   iptables -X  However after a quick check at yougetsignal.com, my port 25 is still being blocked. But 143, 80 are okay. I'm looking for a library in Python which will provide at and cron like functionality.  I'd quite like have a pure Python solution, rather than relying on tools installed on the box; this way I run on machines with no cron.  For those unfamiliar with cron: you can schedule tasks based upon an expression like:    0 2 * * 7 /usr/bin/run-backup # run the backups at 0200 on Every Sunday  0 9-17/2 * * 1-5 /usr/bin/purge-temps # run the purge temps command, every 2 hours between 9am and 5pm on Mondays to Fridays.   The cron time expression syntax is less important, but I would like to have something with this sort of flexibility.   If there isn't something that does this for me out-the-box, any suggestions for the building blocks to make something like this would be gratefully received.  Edit I'm not interested in launching processes, just "jobs" also written in Python - python functions. By necessity I think this would be a different thread, but not in a different process.  To this end, I'm looking for the expressivity of the cron time expression, but in Python.   Cron has been around for years, but I'm trying to be as portable as possible. I cannot rely on its presence. Size of files in Windows OS. (It's KB or kB?) Is it OK to require users to have a third party login account? Is there a way to force Google Photos to recognize new faces? I have "Face grouping" enabled in my Google account and I can see around 10 faces successfully tagged when I go to my Albums. However, there's a lot of other people who are not tagged, as well as some unrecognized photos of people who are already tagged.   Is there a way to force Google Photos to run the facial recognition algorithm on a given photo, so that it adds new faces or maps a new photo to an existing face? Due to the implementation of Java generics, you can't have code like this:   public class GenSet&lt;E&gt; {     private E a[];      public GenSet() {         a = new E[INITIAL_ARRAY_LENGTH]; // error: generic array creation     } }   How can I implement this while maintaining type safety?  I saw a solution on the Java forums that goes like this:  import java.lang.reflect.Array;  class Stack&lt;T&gt; {     public Stack(Class&lt;T&gt; clazz, int capacity) {         array = (T[])Array.newInstance(clazz, capacity);     }      private final T[] array; }   But I really don't get what's going on. I found the following proof arguing for the irrationality of $\sqrt{2}$.     Suppose for the sake of contradiction that $\sqrt{2}$ is rational, and choose the least integer $q &gt; 0$ such that $(\sqrt{2} - 1)q$ is a nonnegative integer. Let $q'::= (\sqrt{2} - 1)q$. Clearly $0 &lt; q' &lt; q$. But an easy computation shows that $(\sqrt{2} - 1)q'$ is a nonnegative integer, contradicting the minimality of $q$.   Clearly, this is a vague proof in that it expects the reader to work out the math involved on his own. I took that as an exercise and worked out the following:  Assuming $\sqrt{2}$ to be rational, let $$\sqrt{2} = \frac{p}{q}$$ where $p$ and $q$ are least possible integers.  Clearly then, $\sqrt{2} - 1$ would also be rational, $$\sqrt{2} - 1 = \frac{p-q}{q}$$ where both $(p-q)$ and $q$ are also integers.  Multiplying both sides by $q$, we get $$(\sqrt{2} - 1)q = (p-q) = q'$$ where $q'$ is an integer. Using, $2&gt;\sqrt{2} &gt; 1$ , we get $$1&gt;\sqrt{2}-1&gt;0$$ proving that $q'$ is a non-negative integer.  As stated in the proof, computing $(\sqrt{2}-1)q'$ we get, $$(\sqrt{2}-1)q' = \frac{(p-q)^2}{q}$$which is not necessarily an integer (in fact necessarily not an integer). However, the proof claims it to be.  Am I missing something or something is wrong with how I proceeded?  Here is a link to the textbook where I found this proof: , (p 32, problem 1.14).  P.S. This is my first post here and my first time with TeX. I'd also like to know if there is anything wrong with the way my question is presented. Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$. Let $A$ be an $n \times n$ matrix.  $i)$Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.  $ii)$Prove that if the sum of each column of $A$ equals $s$, then $s$ is an eigenvalue of $A$.  I think that being an eigenvalue of $A$ implies that $sv=Av$ for some vector $v$. Furthermore, I know that $[a_i] = s$ if we let $a_i$ denote the i-th row of $A$. However, I do not seem to be able to find a link between these two facts. Could anyone please help me out? what is the difference between "yet" and "still"?   When we can use "still"? and when we can use "yet"?   Are they synonyms or not?  "The plan could yet succeed." Can I use "still" in that sentence instead of "yet"?  "I still don't understand." Can I use "yet" in that sentence too? Why Java needs Serializable interface? We work heavily with serialization and having to specify Serializable tag on every object we use is kind of a burden. Especially when it's a 3rd-party class that we can't really change.  The question is: since Serializable is an empty interface and Java provides robust serialization once you add  implements Serializable - why didn't they make everything serializable and that's it?    What am I missing? Do mindless creatures get feats? Do creatures that have no Intelligence Score ("Int —" in the stat block) gain Feats? I'm creating some monsters but I failed to find the answer for this question. How are qubits better than classical bit if they collapse to a classical state after measurement? Classical computers store information in bits, which can either be $0$ or $1$, but, in a quantum computer, the qubit can store $0$, $1$ or a state that is the superposition of these two states. Now, when we make a "measurement" (to determine the state of qubit), it changes the state of the qubit, collapsing it from its superposition of $0$ and $1$ to the specific state consistent with the measurement result.  If from a measurement of a qubit we obtain only a single bit of information ($0$ or $1$, because the qubit collapses to either of the states), then, how is a qubit better than a classical bit? Since ultimately we are only able to get either $0$ or $1$ from either classical or quantum bit.  If you can explain with an example, it may be helpful. I just spent the last 10 minutes trying to put a new set of grips on to my mountain bike handlebars..  The old ones came off fairly agreeably and I've got the new ones 80-90% of the way on, however, at this point both my hands are stinging from the aggressive twisting and pushing - I have tried gloves, but I feel it's best left till my hands don't hurt anymore.  Is there a tried and tested way of doing this? Or is there a knack to this that I am missing?