How to prove that $1^r + 2^r +... + n^r = a_1 n^1 + a_2 n^2 +... + a_{r+1} n^{r+1}$? I see this equation  $$ 1^r + 2^r +... + n^r = a_1 n^1 + a_2 n^2 +... + a_{r+1} n^{r+1} $$ in Introduction to Linear Algebra, where $a_k$s are some constants.     How can I prove it?
Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula As far as every question I've seen concerning "what is $\sum_{k=1}^nk^p$" is always answered with "Faulhaber's formula" and that is just about the only answer.  In an attempt to make more interesting answers, I ask that this question concern the problem of "Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula for fixed $p\in\mathbb N$".  I've even checked this  without finding what I want.  Rule #1: Any method to compute the sum in question for arbitrary $p$ is good, either recursively or in some manner that is not in itself a closed form solution.  Even algorithms will suffice.  Rule #2: I don't want answers confined to "only some values of $p$".  (A good challenge I have on the side is a generalized geometric proof, as that I have not yet seen)  Exception: If your answer does not generalize to arbitrary $p$, but it still generalizes to an infinite amount of special $p$'s, that is acceptable.  Preferably, the method is to be easily applied, unique, and interesting.  To start us off, I have given my answer below and I hope you all enjoy.
Recommended shutter speed for action sports? What shutter speed do professional sports photographers typically use for action sports like football and basketball? Intuition behind the definition of a measurable set How to extend my root (/) partition? Till today I was using ubuntu 13.04 in my windows7/ubuntu dual boot. Just now I have upgraded to ubuntu 14.04. My software updater showed an updates available message. I proceeded with  Install Now . Soon I got the following window.    Then I tried sudo apt-get clean as recommended. Nothing happened! Not even a window opened. Teach me deleting the temporary packages.   Deleting temporary files may not be highly helpful in case of further updates and installs for I don't have too much space left in my /. I wanna extend my / partition with the help of the unallocated space in my HD. The following is my Gparted window.       I got an answer . But it wasn't useful for me. I had two choices. But the answer wasn't working for both my chioces.  Choice 1: I may use the 4.25 GiB unallocated space to extend my /dev/sda10. To do so, I must need to move my linux-swap partition. But my Resize/Move option is disabled for the linux-swap partition. Choice 1 failed!   Choice 2: I may use the 14.35 GiB unallocated space instead. In this case I must move more than one partition. /dev/sda4  partition is the first on the list. Even that partition has the same problem. Resize/Move option is disabled. Choice 2 failed!  I am out of choices now.  What shall I do? Help me to extend my / with Gparted. Is there a way in Java to have a map where the type parameter of a value is tied to the type parameter of a key? What I want to write is something like the following:  public class Foo {     // This declaration won't compile - what should it be?     private static Map&lt;Class&lt;T&gt;, T&gt; defaultValues;      // These two methods are just fine     public static &lt;T&gt; void setDefaultValue(Class&lt;T&gt; clazz, T value) {         defaultValues.put(clazz, value);     }      public static &lt;T&gt; T getDefaultValue(Class&lt;T&gt; clazz) {         return defaultValues.get(clazz);     } }   That is, I can store any default value against a Class object, provided the value's type matches that of the Class object. I don't see why this shouldn't be allowed since I can ensure when setting/getting values that the types are correct.  EDIT: Thanks to cletus for his answer. I don't actually need the type parameters on the map itself since I can ensure consistency in the methods which get/set values, even if it means using some slightly ugly casts. Let $A$ be a commutative ring, $I$ an ideal of $A$ and $M$ an module over $A$. Is it true that $\operatorname{Ann}(M/IM) = \operatorname{Ann}(M) + I$?    One inclusion is certainly true, but I don't know about the other one. If the above statement does not hold in general, does it perhaps for finitely generated modules? Singular or plural "set"? C# compiler error: "cannot have instance field initializers in structs" What do you call a person who is your friend only because you benefit them? Looking for a word to describe a person who only makes friends to benefit themselves. Runtime error : ERROR 000358: Invalid expression using SelectLayerByAttribute in ArcPy? I cannot for the life of me persuade SelectLayerByAttribute to work (I need to automate this for a complex script).  Working in ArcMap's python console on already loaded shapefile, the format below:  import arcpy from arcpy import env arcpy.env.workspace = r"C:/Users/.../working" arcpy.SelectLayerByAttribute_management("layer", "NEW_SELECTION", " 'ATTRIBUTE' = 'Criterion' ")   results in error:     Runtime error : ERROR 000358:   Invalid expression Failed to execute (SelectLayerByAttribute).   An alternative method:  import arcgisscripting import arcpy gp = arcgisscripting.create(9.3) gp.workspace = "C://...//working" sFile = "C://...//working//shapefile.shp" lyr = "layer" gp.makefeatureLayer_management(sFile, lyr) gp.toolbox = "management" gp.SelectLayerByAttribute("layer", "NEW_SELECTION", " 'ATTRIBUTE' = 'Criterion' ")   .. selects everything, including features that don't meet the criterion!  Does anyone have any idea what's going wrong?  I've read some pages on , but found no solution that works here. The working directory is correct. The attribute field and criterion are correctly specified (case sensitive etc), etc. Its very confusing. Why would I want to install a snap if I can install via apt instead? Limit $\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$ How can I find the following limit :$$\lim_{n\to\infty}\left(1-\frac{1}{3}\right)^2\left(1-\frac{1}{6}\right)^2\left(1-\frac{1}{10}\right)^2\cdots\left(1-\frac{1}{\frac{n(n+1)}{2}}\right)^2$$There are four options $(a)\frac1{3}(b)\frac1{9}(c)\frac1{81}(d)0$ Thanks! How can I automatically add some skeleton code when creating a new file with vim When creating a new file with vim, I would like to automatically add some skeleton code.  For example, when creating a new xml file, I would like to add the first line:    &lt;?xml version="1.0"?&gt;   Or when creating an html file, I would like to add:  &lt;!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"&gt; &lt;html&gt;   &lt;head&gt;     &lt;title&gt;&lt;/title&gt;   &lt;/head&gt;   &lt;body&gt;   &lt;/body&gt; &lt;/html&gt; find the number of analytic functions which vanish only on $S$? Standard nomenclature for component types What prefixes should be used on reference designators for components of various types?  I think we can all agree that "R##" refers to a resistor, "C##" refers to a capacitor, and "L##" refers to an inductor.  Beyond that, it appears to be a mishmash of conventions.    I've seen both "IC" and "U" used for chips.   "Q", "T", and "M" have been used for transistors and MOSFETs.  Ordinarily, "D##" is used for diodes, but LEDs and Zeners get special treatment on some boards.    What standards are available for reference, and who endorses them?  JEDEC, ISO, IEEE, and other standardization bodies are welcome references (though I'd prefer an inexpensive standard), but I'm also curious to see what's used at various companies other than my own.    Edit:  What I'd really like to see is a list in an answer here which complies with the standard (even if it's just a quote). Huge paged pool probably caused by fltMgr.sys , what to do? How can I install Sun/Oracle's proprietary Java JDK 6/7/8 or JRE? I want to install Oracle's JRE and to update to the latest version with the Software Updater when they released. Is there a Ubuntu package that is provided by Canonical or Oracle?  Before release Java 7, I followed  to install Java 6.  But it doesn't work for Java 7. There is no package sun-java7-xxx. How can you install Java 7? Highlight a word with jQuery I basically need to highlight a particular word in a block of text. For example, pretend I wanted to highlight the word &quot;dolor&quot; in this text: &lt;p&gt;     Lorem ipsum dolor sit amet, consectetuer adipiscing elit. &lt;/p&gt; &lt;p&gt;     Quisque bibendum sem ut lacus. Integer dolor ullamcorper libero.     Aliquam rhoncus eros at augue. Suspendisse vitae mauris. &lt;/p&gt;  How do I convert the above to something like this: &lt;p&gt;     Lorem ipsum &lt;span class=&quot;myClass&quot;&gt;dolor&lt;/span&gt; sit amet, consectetuer adipiscing elit. &lt;/p&gt; &lt;p&gt;     Quisque bibendum sem ut lacus. Integer &lt;span class=&quot;myClass&quot;&gt;dolor&lt;/span&gt; ullamcorper     libero. Aliquam rhoncus eros at augue. Suspendisse vitae mauris. &lt;/p&gt;  Is this possible with jQuery? Edit: As  , this is quite possible without jQuery - but I was hoping there might be a special method of jQuery which would let you do selectors on the text itself. I'm already using jQuery heavily on this site, so keeping everything wrapped up in jQuery would make things perhaps a bit more tidy. Just posted a bug report, and found another one :p    (lame hand-drawn cursor coz windows doesn't capture the real cursor)  Oh yeah, I'm on Firefox. How to switch from Resident visa to Tourist visa status in the Schengen area? How to return to original .pdf presentation after open a .pdf linked file? I'm preparing a beamer presentation and I'm using hyperlinks for the first time; When I do something like:  \href{name_file_to_open.pdf}{here I put a link to a figure which is working well}   It works just fine and open the .pdf document; Problem is I can't return to my presentation, cause when I close the linked.pdf I close it all. (I try this with movie and indicating an area to the file open like \href[width=3cm,height=2cm] but then it doesn't work)  Can anyone please has any suggestion to this problem?