Im studying a book about stochastic processes and it is mentioned in the book that the equation below is true and we can obtain it by generating functions can anyone prove it. Thanks a lot. $$\sum_{m=0}^{\infty}\binom{2m}{m}p^m = (1 - 4p)^{-1/2}$$
How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in studying the first passage time of a simple random walk.)
It maby be weird, I want to make header and footer for every page including the first page of each chapter, how can I do this? When is unbalanced data really a problem in Machine Learning? I have multiple cameras in a scene. What I want to do is to change which one is active. This way I can view the scene from different angles. How would I do this? Can Ubuntu use all my 4GB of RAM? Windows update doesn't progress beyond 0 KB 0% I have Windows&nbsp;7 64-bit and the  isn't working properly. What can I do? Access Java / Servlet / JSP / JSTL / EL variables in JavaScript I try to launch Firefox over SSH, using  ssh -X user@hostname   and then  firefox -no-remote   but it's very very slow.  How can I fix this? Is it a connection problem? This may be a simple question, but how do I use files starting with "-" with command line programs?   For example,  I am trying to use pdfgrep on a file named -2013-01-01.pdf but it complains that there are no options defined for 2, 0, 1, 3 and so on.  Changing the filename to something that does not start with "-" solves it, but this is not an option since the files aren't created by me. I simply want to check for a specific change I know is coming. Make terminal use pageup/pagedown instead of shift-pageup/shift-pagedown? I've grown quite weary of shift-pgup to page up in gnome terminal, the command line, terminator, and guake, etc.  Any way to easily make these apps use the straight page-up and page-down keys for paging around? Is it possible with GEE to export all the images in an image collection? Related to the question :  It is well known that   $$\zeta(2n) = f(2n) \pi^{2n}$$  where $f(n)$ is an function in rationals: (the denominator = OEIS : 6, 90, 945, 9450).  Apery proved that $\zeta(3)$ is irrational, and presumably $f(2n+1)$ is so for all (but the latter is open).  But what else is known about $f(2n+1)$? What sort of pattern does it share (if any) with $f(2n)$? Numerically does it fit well? Is there any clue/progress since Apery as to its form? Of course, $f(2n+1)$ may in effect divide out the $\pi$, meaning that somehow $\pi$ is not involved in $\zeta(2n+1)$ in any essential way. Debugging a custom-bib bst file that crashes on book items I created my own bibliography file with custom-bib (makebst), but when I run it with a reference to a book I get:  $ bibtex PhD [...normal output] Database file #1: PhD.bib You can't pop an empty literal stack for entry oecd2001brc while executing---line 1504 of file wdsphd.bst   The only thing on that line is a high-level ITERATE call.   The style was created with these options:  %% merlin.mbs  (with options: `head,exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}') %% english.mbs  (with options: `exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}') %% merlin.mbs  (with options: `tail,exlang,annote,seq-no,nm-rvvc,dt-beg,yr-par,xmth,tit-qq,qx,bt-qq,vol-bf,volp-sp,num-xser,pub-date,isbn,issn,doi,english,pp,ord,url,url-nl,nfss,,{}')   The code for book items looks like:  FUNCTION {book} { output.bibitem   author empty$     { format.editors "author and editor" output.check     }     { format.authors output.nonnull       crossref missing$         { "author and editor" editor either.or.check }         'skip$       if$     }   if$   format.btitle "title" output.check   crossref missing$     { format.bvolume output       new.block       format.number.series output       new.sentence       format.publisher.address output     }     {       new.block       format.book.crossref output.nonnull     }   if$   format.edition output   format.isbn output   format.doi output   new.block   format.note output   fin.entry   write.url }   And this is the entry:  @book{oecd2001brc,  author = {OECD},  doi = {10.1787/9789264193550-en},  isbn = {9789264186903},  publisher = {OECD Publishing},  title = {{Biological Resource Centres: Underpinning the future of life sciences and biotechnology}},  url = {http://www.oecd-ilibrary.org/science-and-technology/biological-resource-centres\_9789264193550-en},  year = {2001} }   Is there a way to get a better stacktrace from bibtex? What I have now doesn 't really tell me what is going on. If it can be fixed simply, help would also be appreciated. My Ubuntu 12.04 was normally updated through the Update Manager, something about gtk. Then I rebooted and now all that shows after the splash screen is a blue screen that only shows the cursor. I need a counsel about my websites' domain.  Let's say I have just bought two domains: xyz.com and xyzphotography.com.  As we can see, the first one is shorter but way less meaningful than the second one. The second one contains a photography keyword but it's longer.  I have only one website which is my photography portfolio. Can you advise me how to set those domains to make it all as SEO-friendly as possible? Which one should be the main one, does the rest should work on their own or have some kind of SEO-friendly redirect to the main domain? Creating intuition about Laplace & Fourier transforms Bring back review numbers to the top bar Though I never quite understood what those numbers were and they were quite ugly, they were the only reason I ever visited the reviews page.   Can we have similar functionality back on the top bar please? How do I check if an element is hidden in jQuery? What does role db_owner allow I have been trying to troubleshoot a login that is not able to view certain tables in a SQL Server 2012 database. In doing so I find I don't quite understand what membership in the db_owner role allows. I can understand the other roles such as db_datareader and db_datawriter but I remain confused as to what db_owner allows. A lot of my scripts stopped working after installing Ubuntu 15.10 and I have trouble remembering this new name as I was used to ethx and wlanx for years. Is there a reason for this change? Do I have to get used to this or can I just rename it and go back to good old eth0?  $ ifconfig  enp0s25   Link encap:Ethernet  HWaddr 77:5a:5e:a6:86:d5         inet addr:192.168.31.239  Bcast:192.168.31.255  Mask:255.255.255.0       inet6 addr: fe80::725a:b6ff:fea6:86d7/64 Scope:Link       UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1       RX packets:4833 errors:1332 dropped:0 overruns:0 frame:666       TX packets:5589 errors:0 dropped:0 overruns:0 carrier:0       collisions:0 txqueuelen:1000        RX bytes:3266446 (3.2 MB)  TX bytes:1046654 (1.0 MB)       Interrupt:20 Memory:d3500000-d3520000 Seeking QGIS tutorials and web resources? I wish to familiarize myself with the capabilities of QGIS up to and beyond QGIS version 3.0.   Can you recommend some good web resources and tutorials that would show me QGIS' capabilities?   I am interested in both using QGIS through its GUI and in developing applications using QGIS.