I'm trying to compute the infinite sum  $\sum_{n=1}^{\infty}n(\frac{1}{2})^n$  which I believe should represent the expected amount of coin flips needed to get a head. Can someone remind me how to do this?
Apparently,   $$ \sum_{n = 0}^\infty \frac{n}{2^n} $$  converges to 2. I'm trying to figure out why. I've tried viewing it as a geometric series, but it's not quite a geometric series since the numerator increases by 1 every term.
How To: Execute command line in C#, get STD OUT results How do I execute a command-line program from C# and get back the STD OUT results? Specifically, I want to execute DIFF on two files that are programmatically selected and write the results to a text box. How to allow non-superusers to mount any filesystem? Is it possible to allow some particular users (e.g. members of a group) to mount any filesystem without superuser privileges on Linux?  Another question might have been "in what ways a user can harm a system by mounting filesystems?" Create a .desktop file that opens and execute a command in a terminal Does anyone know of a nice (and preferably free) way to integrate Prolog and C#?  Im looking to create a Prolog dll or similar to call from my managed code, and retrieve an answer once all the processing has been complete. Im looking for it to be predominantly one sided (c# calls Prolog).  I have seen  which talks about Prologs real world usage but I was wondering if anyone had either any experience with c# &amp; Prolog? or a nice tutorial/article? How to determine the maximum number to pass to make -j option? $\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in... Let $\{a_n\}$ be a sequence of positive real numbers such that   $a_1 =1,\ \  a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$.   Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in...  (A) $(1,2]$, (B) $(2,3]$, (C) $(3,4]$, (D)$(4,5]$.   Solution attempt:   Firstly, we figure out what $\frac{a_{n+1}}{a_n}$ is going to look like. We get, from the recursive formula, $\frac{a_{n+1}}{a_n}=1+\sqrt{1+\frac{1}{a_n^2}}$ (remembering the fact that $a_n&gt;0$, the other root is rejected).   We know that, if $\lim_{n \to \infty}\frac{a_{n+1}}{a_n}&gt;1$, then $\lim a_n \to \infty$.   Further, $(a_{n+1}-a_n)= \sqrt{a_n(a_n+1)}&gt;0$. (Again, the other root is rejected due to the same reason).   Hence, $(a_n)$ increases monotonically.   Therefore, the largest value of $\frac{a_{n+1}}{a_n}$ is approximately $1+\sqrt{1+\frac{1}{1}} \approx 2.15$  Now, the sum can be approximated as $\displaystyle\frac{\frac{1}{3}}{1-\frac{2.15}{3}} \approx 1.3$ (In actuality, $\mathbb{sum}&lt; 1.3$).  So, option $(A)$ is the correct answer.   Is the procedure correct?   I have been noticing a handful of this type of questions (based on approximations) lately, and the goal is to find out where the sum / the limit of the sequence might lie.   Is there any "definitive" approach that exploits the recursive formula and gives us the value, or does the approach varies from problem to problem? Travel to Australia with new passport (with visa in old passport) I was granted a visa to Australia few months ago. I’d like to travel to Australia in few days. However, I’ve just renewed my passport and I don’t have enough time to inform them about my passport renewal. Can I travel to Australia by carrying both my old and my new passports? Suppose Alice publishes a book with a public key in it, and later wants to prove that she wrote the book. She could sign challenge messages with her private key, and others could verify those signed messages. Alice is proving she knows a private key without revealing anything about it.  Is this an example of an interactive zero knowledge proof?   Or does this example somehow not apply to zero knowledge proofs? This can probably be fixed quite easily. I tried to research and find the answer to this but couldn't find anything relevant.     Basically, I can access my website perfectly through its url. (ex. example.com)  Whenever I click a link, go to my /wp-admin, or go to any other page, my domain changes into the IP address. (ex. instead of showing example.com/about it would show 192.111.223.4/about)  How do I fix this? Do I have to edit a file or can this be fixed by adding a domain record?  Thanks. Hill Cipher known plaintext attack I know a plaintext - ciphertext couple of length 6 for a hill cipher where its key is a [3x3] matrix.  Based on what I've read and learned, to attack and crack keys of [n x n], if we know a plaintext - ciphertext duo of length $n^2$ then we have our set of n equations with n variables, and this is generally solvable. However in my case there is only length of $6$ instead of $n^2 = 9$ .   My question is, how will I solve this problem and find the key? Or, since this is an obvious homework question, what should be my way of thinking? I cannot think of anything else but matrix multiplications and inverses but they do not help me at all. How do I remove old kernel versions to clean up the boot menu? sql server 2008 management studio not checking the syntax of my query 16.04 VirtualBox VM (from vhd file) hangs at "non blocking pool is initialized" I'm trying to boot Canonical's Azure 16.04 .vhd file in a VirtualBox VM and it hangs at "non blocking pool is initialized".  It works in Hyper-V, so it may only be a bug in virtual box (5.0.20). Why multiple PTR records in DNS is not recommended? I   that using multiple PTR records in a DNS configuration is not recommended.   However, the reasons are often vague, or not so obvious, naming:    "it can cause problems",  "can trigger bugs in programs expecting a single answer" : it's the software's problem then, isn't it?! "can make DNS answer packet too large" : isn't this fixed with ?   Are these good reasons? Do you know of any other (good) reasons? All this kinda looks like a "legacy fear"... I've removed the poll style answers in favour of open discussion.  I realise now this isn't the best way to have a proper discussion about this issue - sorry to anyone who commented/voted.    This is mostly a followup on this question:    As I've answered, the reason this is happening is that the root database (in the production Stack Exchange database server) is still called StackExchange.Programmers - a database name that propagates to SEDE and through it to the data dump.  The question here is what to do about it - the current situation is somewhat broken for users of the data dump.  These are the options we have:   Do nothing Update the root database name (requires downtime - so very undesirable) Update the update scripts so the SEDE database name is the new name, change will flow to the data dump export (breaks any queries using the old name) Update the export to the data dump so only it gets the new name Something else we haven't thought about   I have added all of these options as answers below - please vote on the one you most prefer, we will try to accommodate the top voted answer. Like the title said, I want to install Celestia. I already downloaded the packages  and then, I tried all the things that I found on some posts (also, they're outdated, cause the last one is for Ubuntu 16.04) but nothing works.  How can I install it on Ubuntu 18.04.1? Cohen's d for 2x2 interaction I have a significant 2x2 interaction in a between-subjects design.  I would normally report omega-squared as my effect-size measure, but I've been asked to provide Cohen's d instead?  Can Cohen's d be computed for an interaction?  If so, how? Removing a list of characters in string Quick doubt about multiplying vectors by scalars I've just started studying linear algebra, and I'm having trouble understanding something that apparentely seems obvious to everyone else. My question is, why does multiplying a vector by a positive scalar does not change its direction? I've seen some visual representations of this and it does seem reasonable to me that that should be the case. However, I wonder whether there is a way to actually show it mathematically? In the 2d coordinate plane, for instance, I guess way one of showing that vector $A$ has the same direction of $cA,c&gt;0$ would be to show that the line containing $A$ has the same slope as the one containing $cA$ (I'm not sure if this approach is correct though). However, how to proceed in the case of $n$ dimensions? I think I must be missing out something basic here, but I really can't seem to get my head around it. Any help will be highly appreciated. Thanks very much in advance. I want to burn a Windows ISO to a USB device in Ubuntu. How do I do this?   I know how to burn a Ubuntu ISO into a USB device, but with a Windows ISO it's not the same.