While reading slides to one of my courses I came across the following: $\sum_{i=0}^\infty {2i \choose i} a^i$. The original series was a little more complicated but is essentially equivalent to this one. In the slides, the professor had written that the series is equal to $\frac{1}{\sqrt{1 - 4a}}$.  Could you please provide me a hint on how to approach this problem? When I used Stirling's approximation I was able to obtain $1 + \sum_{i=1}^\infty \frac{1}{\sqrt{\pi i}} a^i$, is this the right way? My guess is not, since in the slides there is an equal sign, not an approximation.
I need to find the values of $a$ and $n$ in the following  $$ (1 + ax)^n  = \sum_{i=0}^\infty \binom{2i}{i}x^i$$  How can I compare things and show that $a=-4$ and $n = -{1 \over 2}$. It's easy to expand $1 \over \sqrt{1 -4x}$ see that it's valid but I'm looking other way around.
Is it true that all numbers of the form $\sqrt[b_1]{a_1}+\sqrt[b_2]{a_2}+\cdots+\sqrt[b_k]{a_k}$ are not integers, where $a_i,b_i$ are positive integers such that none of the $k$ terms is an integer? I think this must be a theorem somewhere, probably a fundamental theorem in number theory, but not sure what it would be called. What Happens to electrons after Alpha Decay and Nuclear Fission? Where do the electrons go? In alpha decay do 2 electrons follow the alpha particle and make stable Helium or does the larger daughter nucleus become an anion?  Also what do the electrons do in the mixture of fission and alpha decay? With Beryllium-8 it decays into two alpha particles and 4 lonely electrons or do you get two Helium atoms?If someone could add an entry to Holocron it would be greatly appreciated, thanks. Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$ Switching baselayers as user zooms in to map in OpenLayers? Finding the derivative of $x\uparrow\uparrow n$ I am trying to find a general derivative for the function: $f(x)=x^{x^{x^{...^{x}}}}$however to do that I must find $f^{\prime }$ and $f^{\prime \prime}$...etc. I am now trying to write down a general expression for $f^{\prime \prime}$ and I have stumble opon the series in the picture below, I wonder, do anyone have an idea of what sort of expantion it migth be ? I have tried breaking it down into three different sums, but I wasn't very successful!  EDIT  So since this post has gotten some attention and the title has been change I will tell you what I am on about, although I already have a post just for the question in the title from before. I am as I said earlier trying to find a general derivative for the function $f(x)=x^{x^{x^{...^{x}}}}$ where $n$ is an arbitrary number, natural number and greater or equal to 2. I have already made success when I tried to find the first derivative, and I have also found some interesting pattern emerge when the derivatives as well as n incresses. But my question here was if anyone knew what this expation might be or look similar to be, since I have yet to be lernt series etc. I did not know that the question needed such background since it's straight a forward question. Identify $P(2014)$ if $P(i)=1/i$ for every positive integer $1\le i\le2013$ Suppose $P(x)$ is a polynomial of degree $2012$ and $P(x) = 1/x$ when $x$ takes the integer values $1\cdots2013$ (inclusive). What is the value of $P(2014)$?   I get $1/1007$ but I'm not sure if it's right, and my method is very inelegant. iOS: Custom permission alert view text How do I label equations without the chapter number? How to connect to Office Online Server in Office365 I just wondering if we can use existing Office Online Server in Office365 for SP2013 on premise. If yes, how this can be achieve?   In SP2013 we normally run Get-SPWOPIBinding to find out the server binding and to connect to Office web App we normally run  New-SPWOPIBinding -ServerName. Is this the same command if I want to connect to the existing new Office Online Server in Office365?  Thanks i have a table like this  +--------+------------+-----------+ | id     | volume     | price     | +--------+------------+-----------+ |      1 | 1.0000     |  14250.00 | |      2 | 0.2500     |  14251.10 | |      3 | 0.2500     |  14250.00 | |      4 | 0.0050     |  14200.00 | |      5 | 0.5000     |  14255.01 | +--------+------------+-----------+   Simply if volume 1.7500, how to select like bellow by ordering(highest price)  +--------+------------+-----------+ | id     | volume     | price     | +--------+------------+-----------+ |      5 | 0.5000     |  14255.01 | |      2 | 0.2500     |  14251.10 | |      1 | 1.0000     |  14250.00 | +--------+------------+-----------+   And if volume 1.5000 it also select the least amounts of records needed to calculate to the input volume  +--------+------------+-----------+ | id     | volume     | price     | +--------+------------+-----------+ |      5 | 0.5000     |  14255.01 | |      1 | 1.0000     |  14250.00 | +--------+------------+-----------+   some guys say i cant do this with sql After a reinstall of Ubuntu 12.04 ,it's impossible to install commercial games, e.g Oil Rush. It gives this error:  Failed to fetch https://private-ppa.launchpad.net/commercial-ppa-uploaders/oilrush/ubuntu/dists/oneiric/main/binary-amd64/Packages  The requested URL returned error: 401 Failed to fetch https://private-ppa.launchpad.net/commercial-ppa-uploaders/oilrush/ubuntu/dists/oneiric/main/binary-i386/Packages  The requested URL returned error: 401 Some index files failed to download. They have been ignored, or old ones used instead.   I bought these games in 11.10, but I installed these games in 12.04 before... How can I solve this problem? How to remove accents / diacritic marks from a string in Qt? What are the alternatives for storing client side encrypted data? I want to store client side encrypted data in Drupal 7, so that it is encrypted on the client end, and stored and transmitted in encrypted form. The password is never transmitted or stored on the server. The only modules I have found which do this are  and  but they are for Drupal 6. I need something that is user specific, so that different users can use different keys, rather than a single key or a set of keys used on the server end. Are there any similar modules for Drupal 7? is a psychological phenomenon wherein an individual is convinced that they do not deserve the success that they have achieved despite (perhaps extensive) empirical evidence to the contrary.  Some people with imposter syndrome doubt their own ability level. For me, this usually manifests in the thought:  I've somehow convinced everyone into believing that I'm actually good at this.  followed by the horrifying notion that eventually I'll be found out and my world will collapse. I believe that the imposter syndrome is not uncommon in academia. I'm interested to know how other people here have experienced and dealt with it.  In particular, I'm looking for actual tactics that someone might use to combat imposter syndrome.   Why I asked this question: Beyond the obvious reason that this is something I have struggled with for many years and only recently gotten a handle on myself, there's also some evidence that merely realizing that other people experience imposter syndrome helps fellow imposters. As a grad student, I've managed to meet a fair number of people who are in their own struggles with imposter syndrome; however, the tactics and self-trickery that have helped me don't always speak to them and perhaps someone else's methods might. Changing an element's ID with jQuery I need to change an element's ID using jQuery.    Apparently these don't work:  jQuery(this).prev("li").attr("id")="newid" jQuery(this).prev("li")="newid"   I found out that I can make it happen with the following code:  jQuery(this).prev("li")show(function() {     this.id="newid"; });   But that doesn't seem right to me. There must be a better way, no? Also, in case there isn't, what other method can I use instead of show/hide or other effects? Obviously I don't want to show/hide or affect the element every time, just to change its ID.  (Yep, I'm a jQuery newbie.)  Edit I can't use classes in this case, I must use IDs. Getting a stack overflow exception when declaring a large array The following code is generating a stack overflow error for me  int main(int argc, char* argv[]) {     int sieve[2000000];     return 0; }   How do I get around this? I am using Turbo C++ but would like to keep my code in C  EDIT:  Thanks for the advice.  The code above was only for example, I actually declare the array in a function and not in sub main.  Also, I needed the array to be initialized to zeros, so when I googled malloc, I discovered that calloc was perfect for my purposes.  Malloc/calloc also has the advantage over allocating on the stack of allowing me to declare the size using a variable. What happens if a target for removal becomes ineligible after a card has been cast? When you have a card like  which reads "Exile Target Creature with power 3 or less", as I understand it, it becomes untargetable if the opponent's creature power increases thanks to a pump spell like .  But what if the Disregard is already on the stack and targeting; and they respond with Brute Strength? I'm pretty sure (and my opponent with Brute Strength was certain) that it won't exile the creature, but I don't know which rule states this, and there's no gatherer ruling on the spell. How do I print all lines of a file with duplicate values in a certain column My disk already has 4 primary partitions, how can I install Ubuntu? Is the axiom of choice used in the proof that every open set is the union of basis elements (Mukres, Lemma 13.1)