prove the equation by generating functions
Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$
When reading a stack trace like:  [FormatException: Input string was not in a correct format.]    System.Number.StringToNumber(String str, NumberStyles options, NumberBuffer&amp; number, NumberFormatInfo info, Boolean parseDecimal) +2755599    System.Number.ParseInt32(String s, NumberStyles style, NumberFormatInfo info) +112    System.Convert.ToInt32(String value) +68   What do the +68, +112, etc. numbers mean. I'm guessing that they're offsets to something. If so, what? How do I create silhouettes in a studio environment? The trick to creating silhouettes in the field, as far as I know, is to use spot-metering and to expose for the background. However, this is a bit more difficult in a studio environment. What is the key to creating "artificial" silhouettes?  One solution that I'm considering is placing a bright flash directly behind the subject - would that work? Are there either examples, or a mention of this having happened, of students permanently transferring from one magical school to another in Potterverse?  Canon only please (books/JKR/Pottermore).  Please note that I am only asking about transfers, e.g. you must already be studying in one school. Therefore an example of Malfoys considering sending Draco to Durmstrang is not in scope, since that happened before he attended Hogwarts. Also, guest-studying during Tri-Wizard doesn't count as it wasn't permanent.  Also, if the answer is a "no", please provide evidence (other than "well I read the books before and I don't remember a single case like that", unless you have eidetic memory) from entire canon. I'm trying to recall the name of a book I read, probably published after 1990, where alien AI infects satellites and a barrier is established, isolating earth from space that maybe slows down time inside of it, relative to the passage of time outside.  Mars or the moon is also colonized in an attempt to flee the now real possibility that the sun will become a red giant and wipe out humanity in the not-too-distant future (as perceived on earth), but the colony gets one of the time barriers as well.  I think the name of the novel was short, "The _______" (one word).  The story focused more on the human cost and followed one particular family, one member of which was somehow also partially infected by the alien AI, but I don't remember as much about them. Prove that $(p+q)^m \leq p^m+q^m$ The Public Platform team is continuing to work towards  and we’re here today to announce a new onboarding experience for the review queues. In the  (), we learned that each queues’ instructions could be easily misinterpreted and additional information was difficult to find. Without proper guidance or context, new reviewers would contribute inconsistent reviews and risk having their review privileges suspended. With this release, our goals were to 1) better highlight the review privilege and 2) provide upfront instruction and guidance for successful reviewing. New privilege announcements Community moderation is a significant aspect of Stack Overflow and the Stack Exchange sites. We want users to be aware and familiar with this newly earned privilege, but also celebrate the opportunity to contribute to the community in a unique way.  Many users were discovering the review queues by chance. Once the privilege is earned, a new review queue icon with a red alert would appear in the top-right part of the navigation. We are adding a popover to bring more attention to this easy-to-miss icon. We will also be adding notices for when users earn access to other, higher reputation queues. Additionally, new reviewers will also receive an email notification about the newly earned privilege. They will only get the email if they earn the privilege and don't perform a review within 24 hours (anyone who doesn't want to receive email notifications like this can visit their  and turn off &quot;Tips and Reminders&quot;). We are also showing a popover to users who earned access to review queues in the past, but have never done a review or haven't reviewed in the last 30 days. This popover is only appearing for the next 90 days. The goal is to engage users who may not have interacted with the queues since we made  and . Welcome banner  If a reviewer clicks “Learn more” on the on-page popover or goes to the review queue home page (/review), they’ll be greeted with an onboarding message with general information about the review queues. Informational modals  Queue-specific modals will appear upon page load as you visit each queue. They summarize and highlight important aspects and tips about reviewing in a particular queue. This information can be revisited at any time by clicking “Learn more” under the page header. Feedback As always, please leave your feedback and any bugs you may discover related to this release below this post. We will be monitoring this post for the next few weeks (May 4, 2021). After May 4th, please report any further issues as new questions. How to get data off a completely dead Android phone? Phone worked perfectly. Then one day completely dead, nothing at all. It may have been that "one day" was after some time of laying around, cannot recall.  Model: Android AT&amp;T Samsung Galaxy S3 (SGH-i747 SKU S9255)  Already tried;   Charging phone: no visual output at all, no leds Removing battery while charging: no visual output at all, no leds Removing SIM and SD card: no visual output at all, no leds Attempt to turn on: no visual output at all, no leds   I need to get our family photo's off this phone (stored on internal memory). I care less about the hardware/phone itself.  Thank you for any help you can offer! Reduce Spacing In $$ In math, one often needs to put a letter inside the symbols &lt;&gt;, e.g. to denote a cyclic group &lt;x&gt; generated by some element x. However, plain text displays the symbols &lt; and &gt; as an upside down exclamation point and an upside down question mark, respectively, while math type displays a large space like so:  &lt; x &gt;    Is there some way to reduce this spacing? Launching an Applet fails? Problem with IPv6 sudo apt-get update/upgrade I am using Ubuntu 16.04 Beta 2 at the moment and every time I try to update at home via sudo apt-get update or upgrade trough sudo apt-get upgrade I wont get any feedback. Even installing of anything will have the same result. It will just "hang up" on 0% while fetching data.  $ sudo apt-get update [sudo] password for niclas:  0% [Connecting to archive.ubuntu.com (2001:67c:1560:8001::11)] [Connecting to archive.canonical.com (2001:67c:1360:8c01::1b)]   I have no idea how to solve the problem. I already tried to ping6 archive.ubuntu.com with 100% packet loss while just ping wont have any loss.  I changed the DNS-Server (IPv4 &amp; IPv6) to Google's one no solution either. I need a good explanation (references are a plus) on Python's slice notation.   To me, this notation needs a bit of picking up.   It looks extremely powerful, but I haven't quite got my head around it. macOS app for showing external HDD usage statistics? I'm planning on selling a used external HDD on eBay.  Are there any applications for Mac out there (preferably free) that can show me statistics about the usage history for the drive so that I can let potential buyers know the condition more precisely? If a commutative ring is semiprime and its prime ideals are maximal then it is von Neumann regular (absolutely flat). If a commutative ring is semiprime and its prime ideals are maximal then it is von Neumann regular (absolutely flat).  The converse, although not immediately apparent, can be proven quite easily. But I've been stuck on this one for awhile.  I found that Bourbaki has a question in his treatise on commutative algebra, namely: if all points in Spec(A) are closed (ie. prime ideals are maximal) then A/R (where R is the nilradical) is von Neumann regular (absolutely flat). If this can be proven, the above statement follows immediately since the nilradical of a semiprime ring 0?  Ideas?   Ps: this is my first time posting a question so apologies for any formatting issues. Proving the so-called "Well Ordering Principle" Will I still receive OTA updates after rooting my Nexus 5? I just wanna ask that after rooting Nexus 5 will I be receiving OTA updates by Google ? Or I'll have to install a custom ROM ? URL to load resources from the classpath in Java On the computer upstairs we have an option to make the computer make a sound when we press Caps Lock, Number Lock and Scroll Lock. Is there a way to do this on Ubuntu 12.04? In high Reynolds numbers we have turbulent flow. This is because the inertial forces are much greater than the viscous forces. I understand inertial forces to be actually the fictional forces due to the momentum of the moving fluid. And viscous to be the forces that drag each layers together. But why when inertial forces are too high than the viscous forces, there is turbulence? Why don't the molecules just continue to move laminarly due to their momentum? I can assume the viscous forces are not very strong in high speeds (in high inertial forces) to keep them together to form a laminar flow? But again, why they result in irrational movement after the effect of viscous forces is not anymore considerable? As a mechanical analogue, if I have balls attached together and make them travel linearly, and remove their between them attachment, they can continue to travel linearly. But what happens in fluid molecules that turbulence is created? 'Should've seen it glow' or 'should've seen it glowing'? Which one of the following is the correct one?   I should have seen it glow. I should have seen it glowing.   Or are both correct? Would you parse them please? Can I make `find` return non-0 when no matching files are found?