How can I compute sum of $i 4^i$? How can I compute $$\sum_{i=0}^n i 4^i$$ this equation? What is the way?
Summation by parts (finite calculus) of $\sum\limits_{i=0}^{n}i\cdot a^i$ I'm studying the calculus of finite differences and have read about summation by parts:  $\sum f(x)\Delta g(x) = f(x)g(x) - \sum g(x+h)\Delta f(x)$.  The tutorials I'm using go a bit woolly at the point of introducing this technique: they use examples where they set $\Delta g(x)$ to $2^x$, and don't really show how to use it for anything a bit harder. I'm at a loss for basic examples.  I wondered if it is possible to determine  $S.n = \sum\limits_{i=0}^{n}i\cdot a^i$  using summation by parts?  If so, how is it done?  Thanks in advance.
How to cut the non-overlapping part of 2 shapes in Illustrator CS6 I have 2 shapes (see below) a circle and a shape of the US. I want to remove the part of the US that falls outside the circle. How is this possible?    Thanks Openlayers v4.0.1 support Google Maps Javascript API? I want to know if the latest version of Openlayers (v4.0.1) supports Google Maps as a Tile Layer. I can't find any document ion about this. If Openlayers doesn't support Google Maps, could someone  tell me if there is any way to do this? It is commonly stated that there are four fundamental forces, or interactions, in nature. It is natural to consider which of those is responsible for the normal force we meet in elementary physics. What is the nature of the force exerted on a body by the floor on which it rests?  Of the four forces, the only one that seems relevant is the electromagnetic force, and I always assumed the repelling force in question is ultimately the EM one. The Wikipedia article on  seems to agree:     Molecular and quantum physics show that the electromagnetic force is the fundamental interaction responsible for contact forces.    An alternative explanation is that the force results from Pauli's Exclusion Principle; however, reasonable as it may seem, it doesn't seem to answer the question of what force is actually at play here. The exclusion principle isn't a force, or at least it's not normally described as such.  I've recently come across a rather forcefully presented argument that states that the normal force is a macroscopic force that is not directly reducible to the EM force, or to any of the other forces; rather, it does indeed result from Pauli's Exclusion Principle by way of interaction of electron wave functions. The argument further claims that attempts to correct the Wikipedia article have met with the usual "source needed" objections, and that this topic is either misunderstood by or poorly covered in the standard texts.  (The argument can be found ).   Which way is it? Why does rubbing feet against sheets create a spark? Sometimes when I'm in bed at night, I rub my feet against the sheets and light is created. What exactly is going on?     I'm familiar with static electricity, and the triboelectric effect (creating a charge by rubbing 2 insulators together), and electrostatic discharge (releasing charge by touching a grounded conductor).     I believe that all of that is what is at work here, but I'm slightly confused since in other cases, the triboelectric effect and electrostatic discharge seem to occur as two separate steps, with different substances involved, and here, the two effects seem to be occurring simultaneously, with the same substances.     Can someone help me with a step-by-step explanation of the dynamics of what is going on? How do I convince gdal2tiles to output pngs with transparency? Why would someone use WHERE 1=1 AND &lt;conditions&gt; in a SQL clause (Either SQL obtained through concatenated strings, either view definition)  I've seen somewhere that this would be used to protect against SQL Injection, but it seems very weird.  If there is injection WHERE 1 = 1 AND injected OR 1=1 would have the same result as injected OR 1=1.  Later edit: What about the usage in a view definition?    Thank you for your answers.  Still, I don't understand why would someone use this construction for defining a view, or use it inside a stored procedure.  Take this for example:  CREATE VIEW vTest AS SELECT FROM Table WHERE 1=1 AND table.Field=Value I just wanted to ask whether there is any command which would work on common shells (bash, dash, kornshell)? It is supposed to check if the line variable contains any part of the path.  if [[ $line =~ "$PWD"$ ]] ;then Infinite product with sign changes. For infinite products I know the following theorems:  Theorem I. If $a_n \geq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  Theorem II. If $-1 &lt; a_n \leq 0$ for all $n$ then $\prod (1+a_n)$ converges if and only if $\sum a_n$ converges.  My question is what are necessary and sufficient conditions (possibly including more restrictions than in I and II) for $\prod (1+a_n)$ convergence/divergence as related to  $\sum a_n$ convergence/divergence  if the sequence of terms $a_n$ can switch signs repeatedly but always stay greater than $-1$.  I already know the trivial equivalence of $\prod(1+a_n)$ and $\sum\ln(1+a_n)$. What is the effect of the number of cross-type focus points on sharp focus? How could I retrieve the exact duplicate value from conditionally formatted list that highlight duplicates? I have imported a webpage table in to Google Sheets. It frequently being updated. A series of calculations has been conducted on the sheet named Shawn Of the link spreadsheet.  I have filtered the my desired results into the Calculation sheet of that linked document... which is in columns A5:A. I then applied =ARRAYFORMULA(if(A5:A200=&quot;&quot;,,A5:A200*{12,13,37,101,111})) in columns B5. The Results of the formula extends from B5:F. This results from a5:F was subsequently conditionally formatted to highlighted duplicates in the aforementioned, i.e. a5:f using the lists of values in K5:K and I5:5. In Columns A5:F, is it possible to generate/place values in the range A5:f with matching name of states and aligned values in J5:K and H5:I? I have a 2&nbsp;GB Kingston  card which is about three years old. I put it in a reader today in my Windows Vista computer, wrote a 32&nbsp;MB file onto it, safely removed it, and then tried to read it elsewhere. Nothing. Putting it back in the Windows Vista computer it now says:     You need to format the disk in drive F: before you can use it.   What should I do? I have access to many computers and OSes if your recommendations need that. I would be very sad if I lost all the contents of the card. Most of the data is backed up, but there are a few things that aren't. :(  Doing a   # dd if=/dev/sdg of=~/tmp/sd.bin   gives me a 2&nbsp;GB file, and  the file it seems like lots of my data is still there. How can I put it back together? How can I apply an effect when a player left-clicks/right-clicks with an item in hand? Is there a way so that I make make it when I left click/right click with a certain item it applies an effect? I know that you can apply attributes to wearing armor and things but I want it so when you use the item it applies an effect on your player.  For example, let's say I want something so that when I right click with a certain item (let's say an iron ingot that was given to me) I'll then have a jump boost for 10 seconds. So kinda like a potion in a way.  Is this possible with command blocks/commands/any mods? I just installed a new Ubuntu Server 18.04. I set my hostname hostnamectl set-hostname ****.openbayou.biz and I set /etc/hosts:  127.0.0.1 localhost [ip address] ****.openbayou.biz hostname # The following lines are desirable for IPv6 capable hosts [ip6 address] *****.openbayou.biz hostname ::1     localhost ip6-localhost ip6-loopback ff02::1 ip6-allnodes ff02::2 ip6-allrouters   I also installed OSSEC to monitor for new files, errors and changes to my server and I'm now getting these alerts:  Server returned error NXDOMAIN, mitigating potential DNS violation DVE-2018-  0001, retrying transaction with reduced feature level UDP.`   It's now repeating itself:  systemd-resolved[3195]: message repeated 4 times: [ Server returned error  NXDOMAIN, mitigating potential DNS violation DVE-2018-0001, retrying transaction  with reduced feature level UDP.]   I've looked online for a solution and nobody is reporting this issue. In Unity, are there any pre-defined shortcuts, e.g. to open the dash or unhide the launcher? WPF CommandParameter is NULL first time CanExecute is called What advantages does using (function(window, document, undefined) { ... })(window, document) confer? A word for "intimate friends" without any kind of romance? I'm looking for a word to describe that a particular situation where two people (a man and a woman) are probably the best of friends but have no romantic inclinations towards each other. They may invade each others thoughts periodically but aren't really in a relationship per se. What would be a good word to describe this?  I'm aware of "platonic" but am wondering if there is another word? A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$.  Is there a characterization of the groups in which this property holds?  If this question is too broad, I might ask if such a characterization exists for $p$-groups.    History: I originally posed the opposite question, regarding groups for which $\exists N\unlhd G\,:\, \not\exists H \unlhd G\, \text{  s.t. } H \cong G/N$, and crossposted this to .  I received an answer there to the (now omitted) peripheral question about probability, which shows that most finite groups probably have this property.  After this, I changed the question to its current state, as this smaller collection of groups is more likely to be characterizable. How to avoid interference in wireless communication? I am working on wireless communication system. We are using around 10 pairs of transmitter and receivers. We are using atmega16 microcontroller for encoding and decoding by USART ports.  Now we are able to transmit the data and receive the same in the receiver end, but there is a major problem, when we are finding the 2 transmitter data coming at the same time. The receiver is unable to get it due to interference.  Suppose one transmitter sends "SENDA" at the meanwhile another transmitter sends "GETTS", at that time the receiver can not receive proper data. As all the transmitters and receivers are working in same frequency, so this interference is occurring. How can I resolve this issue? Prove that $f_n$ converges uniformly on $\mathbb{R}$