$$\sum_{k=1}^\infty \mathrm{(1-\frac{1}{k})}^{\mathrm{k}^{2}}$$ I tried using the limit comparison test with $$\sum_{k=1}^\infty \mathrm{(1-\frac{1}{k})}^{\mathrm{k}^{}}$$ but this leads to a limit of 0, which doesn't help. I think this may involve some use of  $\mathrm{e}^x$, but I don't know where else to start. Any suggestions?
Found the alternative form: $\sum _{k=1}^\infty ((1-\frac{1}{k})^{k})^k$. Tried various criteria, no luck so far.
Return multiple values from a SQL Server function How would I return multiple values (say, a number and a string) from a user-defined function in SQL Server? What is a "closed" or "duplicate" question? Questions can be closed or marked as duplicates.  What does it mean for a question to be &quot;closed&quot;? What does it mean for a question to be marked as a &quot;duplicate&quot;? Who can close a question? What are the reasons for closing a question ? Is closure the end of the road for a question? When are closed questions eligible for deletion?   Related        For more information, see &quot;&quot; in the . Implicit Differentiation Help I have to use implicit differentiation to find $\frac{dy}{dx}$ given:  $$x^2 \cos(y) + \sin(2y) = xy$$  I don't even know where to begin, I missed the class where we went over implicit differentiation, and because of that, I am completely stuck.   Thank you everyone.  Edit: I don't know how to make the equation look all nice and whatnot, so sorry about that Is radioactive decay spontaneous or random? When it comes to radioactive decay, what is the difference between random and spontaneous?  For example, when the count rate of a radioactive isotope is measured, the readings fluctuate. Is this a demonstration of the randomness of the process, or of its spontaneous nature? "Overclocking" an AVR In AVR datasheets under the Electrical Characteristics section you will typically find a graph like this (this one is from the ATMega328):    I've seen designs that seem to "work" but operate outside the shaded envelope. Specifically, I've seen 3.3V (Arduino) designs that run the clock from an external 16MHz crystal. Clearly, this is out of spec. What are the practical negative consequences of running outside this envelope? Which edit actions can move a post into Community Wiki mode? Surprisingly, I couldn't find a straight answer to this.  It's not entirely clear in the .  There are (at least?) four different types of edit actions:   Body edits Title edits Retags Rollbacks   And a few actions that show up in the history but aren't really "edits":   Close/Reopen Delete/Undelete   The FAQ seems to indicate that only body edits are important as far as "wikification" is concerned, but I'm not sure if that's really correct or if I'm just interpreting the post too literally.  The FAQ also says that rollbacks don't reverse the calculations - fine, but are they considered to be additional edits or are they ignored?  Can anyone tell me which of these actually count toward the wikification of a post, and whether the rules are any different for owner edits vs. multiple editors? In Windows, one can create a new file using Right Click -> Add New -> Type of File. Is there an analogous action in Lion?  I just want to create a new blank file on my Desktop. Spam iCloud Calendar Invitations Can a Rogue (D&D 3.5) use sneak attack on both the first and second attack? I'm having this Error when I try to compile my .tex file with xelatex:  ! Package hyperref Error: Wrong DVI mode driver option `dvips', (hyperref)                because XeTeX is running.  See the hyperref package documentation for explanation.   This is my LaTeX header  \documentclass[10pt, a4paper, dvips,]{moderncv}  \moderncvtheme[grey, roman]{classic}  \usepackage{ifxetex} \ifxetex % only for xelatex \usepackage{xltxtra} % this should load both fontspec &amp; xunicode \usepackage{fontspec} \setromanfont{Helvetica Neue} \else % \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \fi % \ifxetex   Any ideas?  EDIT:  I removed the 'dvips', now I get:  ! Package hyperref Error: Wrong driver option `pdftex', (hyperref)                because pdfTeX in PDF mode is not detected.  See the hyperref package documentation for explanation.   Now my first latex line  \documentclass[10pt, a4paper]{moderncv}   Any clues ? All matrices which commute with all $2\times 2$ matrices I would like to find all matrices which commute with all $2\times2$ matrices.  I started solving problem in this way:    1) I have this matrix $A$ with real numbers: $$A=\left[\begin{array}{cc}a &amp;b\\c &amp;d\end{array}\right]$$ 2) Matrix which commute with matrix $A$ is matrix $B$: $$B = \left[\begin{array}{cc}e&amp; f\\ g &amp;h\end{array}\right]$$ 3) When i solve the equation $AB=BA$, I get this: $$\left[\begin{array}{cc}ae+bg&amp;af+bh\\ce+dg&amp;cf+dh\end{array}\right]=\left[\begin{array}{cc}ea+cf&amp;eb+df\\ga+ch&amp;gb+dh\end{array}\right]$$  How I can get the general look of wanted matrix? Detecting and acting on keyboard direction keys in Java I'd like to loop though each feature in a layer and calculate its ellipsoidal area using QgsDistanceArea() in PyQGIS 3.  Everything works fine on unprojected layers, but I get strange results when input layer is in projected CRS.  So far I have written this:  from qgis.core import *  def calculate_area(in_lyr_name, ellipsoid, units):     input_layer = QgsProject.instance().mapLayersByName(input_lyr_name)[0]      area = QgsDistanceArea()     area.setEllipsoid(ellipsoid)      features = input_layer.getFeatures()     feature_area = 0     error_features_counter = 0      for i, feat in enumerate(features):         geom = feat.geometry()         polygon_area = 0         try:             if geom.isMultipart():                 polygons = geom.asMultiPolygon()                 for polygon in polygons:                     polygon_area += area.measurePolygon(polygon[0])             else:                 polygon = geom.asPolygon()                 polygon_area = area.measurePolygon(polygon[0])              feature_area += polygon_area          except AttributeError:  # catches NoneType geometries (broken etc.)             error_features_counter += 1             pass      # calculated area is in sq. metres (see the "else" case)     if units == 'km²':         final_area = feature_area / 1e6     elif units == 'Ha':         final_area = feature_area / 10000     else:         final_area = feature_area     return final area, error_features_counter   How do I get a constant result that does not depend on the presence of projection? What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is arbitrary. So saying $\sqrt{-1} = i$ is plainly false!  So why do people say $\sqrt{-1} = i$? Is this how it's taught in the US? I don't see the statistics information that's supposed to be on the status bar running along the bottom of the main window towards the right in version 2.9. Facebook message notification showing blank message Since I updated the system to 10.11.5, the Facebook message notification is not showing message text.   Is this a bug in this software upgrade of OSX? My name includes space - How do I enter my username when typing command? My username in Minecraft is sun solar arrow. This name includes spaces. Situation - When I want to get OP privilege, a user who has OP privilege will enter: /op sun solar arrow  But my username includes spaces. So it shows an error message:  Invaild syntax  How do I enter my username? Is it possible to do something like this  #ifdef SOMETHING #define foo // #else #define foo MyFunction #endif   The idea is that if SOMETHING is defined, then calls to foo(...) become comments (or something that doesn't get evaluated or compiled), otherwise it becomes a call to MyFunction.  I've seen __noop used, but I don't believe I can use that.  EDIT(s):  I don't think I can really use a macro here, because MyFunction takes a variable number of arguments.  Also, I'd like to make it so the arguments are NOT evaluated!  (So doing something like commenting out the body of MyFunction doesn't really give me what I need, as the arguments will still be evaluated) Service or services? I've read many articles that use the term 'emergency services ' and ' location services' .   Why not service instead? Why must it be plural? Thank you A very easy probability problem but opinions diverge greatly. Need some help. The question itself is a very easy one:     Somebody has got two kids, one of whom is a girl. Then what's the probability that he's got at least one boy?   My answer is that, since he's already got a girl, then "he's got at least one boy" amounts to "the other kid is a boy", whose probability is apparently $\frac{1}{2}$. But my friends argue that the probability should be $\frac23$: they say this is a binomial distribution, all the possible cases are (girl,girl),(girl,boy),(boy,girl) which yields that the probability is two cases out of three and is thus $\frac23$. But I think this is totally unacceptable. I don't think it is a binomial distribution at all, at least not what my friends explained to me. However, I just can't disuade them of their opinion, nor can I prove that I am wrong. So what on earth is the probability? and why? Any help is appreciated. Thanks in advance. Esp. Can anybody show why my explanation is wrong? Isn't it that whether the other kid is a boy or a girl a 50/50 event?  EDIT: Thanks for all the help you provided for me, and special thanks will go to @HammyTheGreek and @KSmarts, who have made it clear to me that there is in fact some ambiguity in my statement in this problem. As is pointed in  ,two distinct interpretation of the statement "one of whom is a girl" that gives rise to ambiguity:     From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.   From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of 1/2.