I have been researching the harmonic series, $H_k$ and the Euler-Mascheroni constant, $\gamma$, quite a bit recently. I understand that the diagamma function has this property: $$\psi^{0}(k+1) + \gamma = H_k$$  And equivalently, $$\frac{d}{dk}[ln(k!)] + \gamma = H_k$$ My question is, does there exist some $f(k)$ such that  $$f(k) + \gamma = H_k$$ where $f(k)$ does not require the use of special functions like $\psi$? This question is motivated by the fact that other sums have a formula for the partial sums, for example $$1+2+3+...+n=\frac{n(n+1)}{2}$$ I see how the sum of natural numbers up to $n$ is different than $H_n$; the former always has solutions in $\mathbb{N}$ and the latter (excluding $H_1$) always has solutions in $\mathbb{Q}$. Can $H_n$ have a coinciding formula only using elementary functions, and if not, why?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq n} \frac{1}{k}$, and failed. But should we necessarily fail?      More precisely, is it known that $H_n$ cannot be written in terms of the elementary functions, say, the rational functions, $\exp(x)$ and $\ln x$? If so, how is such a theorem proved?    Note. When I started writing the question, I was going to ask if it is known that the harmonic function cannot be represented simply as a rational function? But this is easy to see, since $H_n$ grows like $\ln n+O(1)$, whereas no rational function grows logarithmically.  Added note: This  asks a similar question for “elementary integration”. I guess I am asking if there is an analogous theory of “elementary summation”.
Sharing a Bluetooth Keyboard and Mouse sequentially between two Macs Why don't more mainstream statically typed languages support function/method overloading by return type?  I can't think of any that do.  It seems no less useful or reasonable than supporting overload by parameter type.  How come it's so much less popular? Using the definition of Riemann integrability, prove that $f $is Riemann integrable on $[0,1]$ The following problem came up on a test, and I didn't get it correctly. Please help me understand the problem.  Problem: Define $f:[0,1] \to \mathbb{R}$ by $$     f(x)=  \begin{cases}     1,&amp; \text{if } x=\frac{1}{n}\\     0              &amp; \text{otherwise} \end{cases} $$ for some integer n.  Using the definition of Riemann integrability, prove that f is Riemann integrable on $[0,1]$ and evaluate the Riemann integral $\int_{0}^{1} f(x)dx$.  My Solution:  Let $\epsilon&gt;0$ and $P$ be the partition of $[0,1]$ such that $P=\{0,\frac{1}{n}- \frac{\epsilon}{3},\frac{1}{n}+ \frac{\epsilon}{3},1\}$.Then note that   $U(f,P)=M_0\Delta x_0 +M_1\Delta x_1+M_2\Delta x_2=0+(\frac{1}{n}+ \frac{\epsilon}{3})-(\frac{1}{n}+ \frac{\epsilon}{3})+0=\frac{2\epsilon}{3}$ and   $L(f,P)=0$.   Thus, for every $\epsilon&gt;0$ $$U(f,P)-L(f,P)=\frac{2\epsilon}{3}&lt;\epsilon$$.  This is the prove that I gave and the mistake that I made was that I fixed n. Some important things that I noticed about this problem were that if you graph the function $f(x)$ most of the points fall to the left or close to zero and a fewer fall to the right. Hence, this makes me believe that I need a partition such that I capture most of the points near zero so that I am only left with a finite number of points close to 1. However, I can't seem to find this partition.Any help would be appreciated. In particular, I would like to know how you come up with the partition since I always struggle to come up with one. Thank you! homemade chilli oil I am making chilli oil using light olive oil, crushed chilli, dried garlic, dried fennel seeds and parsley.  I have been making this for years. I am concerned about this Botulinum business. Should I be concerned? And what can i do to avoid this?  Currently, I heat the olive oil before putting it into the mixture, jar it, and normally keep it in the cupboard for months, with no problems with taste.  Another thing I am wondering is, how does one measure the expiry date on homemade chilli oil? Can someone help? I am trying to restore a database with the following command:  $ sudo -u postgres pg_restore -C -d dvdrental dvdrental.tar  [sudo] password for t:    However, I am receiving the following error message:      could not change directory to "/home/t/mydata/.../relation model/SQL/implementations/Implementations, i.e. relational database management systems/postgreSQL/general/web/postgresqltutorial/databases": Permission denied      pg_restore: [archiver] could not open input file "dvdrental.tar": No such file or directory   I was wondering why I can't change directory to the current directory with permission denied?  File permission bits are:   -rw-rw-r-- 1 t t 2838016 May 26  2013 dvdrental.tar   Is it because one of its ancestry directory is not both readable and executable by any one?  The file has  many ancestry directories, and how can I verify that? meaning of powers on trig functions I always forget this, when a trig function has an exponent does that mean multiply itself or apply itself to the result recursivly?  e.g. does $\sin(x)^2=\sin(x)\sin(x)$ or $=\sin(\sin(x))$? What about $\sin^2x$? I'm putting some type on a path, a circle; but when I do it, the type is upside down. How can I flip it so it reads the right way up? See image below: If I run history, I can see my latest executed commands.  But if I do tail -f $HISTFILE or tail -f ~/.bash_history, they do not get listed.  Does the file get locked, is there a temporary location or something similar? Getting random row through SQLAlchemy Understanding how references and labels work I'm trying to understand why I cannot effectively reference figures or equations that I have defined with labels.  For example:  \begin{figure}   \label{fig:area_Treasuries}   \caption{Treasuries as a portion of total financial flows.}   \centering   \includegraphics[width=0.75\textwidth]{graphs/Area_Treasuries.png} \end{figure}  \paragraph{} Figure~\ref{fig:area_Treasuries} presents Treasuries (both flowing into and out of the United States) as a share of total financial flows.  From 1985 to 2010, Treasuries account for, on average, 53.2\% of total financial flows.   With the above code, I get this warning:      LaTeX Warning: Reference `fig:area_Treasuries' on page 37 undefined on   input line 477.   The result is "Figure ?? presents Treasuries..."  Why is this happening?  Related, I don't understand why I can't reference, say, an equation before (I mean physically above in the text) the actual equation.  Isn't this what compiling multiple times and cross references is supposed to take care of? Why disable Fast Boot on Windows 8 when having dual booting? Narrow-phase collision detection algorithms Inaccurate reputation graphs - bug or as designed? I have notice two issues with the reputation graph: 1: This user has a higher score on the graph than in reality I happened to notice  where the user's graph reputation does not match their actual reputation.  The person in question was hit by a -100 penalty for an offensive answer a while back, but the graph doesn't reflect that.  I thought it just didn't update when I noticed this at the time, but the person has since gained and lost reputation and at all times the graph was +100 of their actual reputation.  It seems, based on Shadow Wizard's comment that this is because the reputation change is non-public. OK, fair enough... , which was marked as , noticed a similar phenomenon for placed DVs.  This prompted me to examine graphs more carefully.  It seems that this hidden -1 has been updated, as I checked several users and did not find any others that had lower actual reputation than graph reputation, so .  (Since the number of DVs placed is public, I can be reasonable sure that these users have placed -1 rep DVs, although strictly speaking they could be 100% question DVs.) 2: Some users have a lower score on the graph than in reality I did, however, find many users where the graph says their reputation is much lower than it actually is.  Perhaps that is from deleted downvotes turning into hidden +1s, but those should be balanced by hidden -1s, so its not clear what happened.  Seems there is some error in the logic - maybe the -1s do affect the graph now as per the , but the corresponding +1s don't?  Whatever the case, I can't imagine it is intentional. Edit: I went back to find some examples today &amp; didn't find any.  I think the issue has been resolved - either it was a caching issue or a change was made to the code after this post. dpkg error: "trying to overwrite file, which is also in..." For example:  $ sudo apt-get install curl Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: bsh : Depends: libjline-java but it is not going to be installed groovy : Depends: libjline-java but it is not going to be installed rhino : Depends: libjline-java but it is not going to be installed E: Unmet dependencies.   Try 'apt-get -f install' with no packages (or specify a solution).   I get the same or similar errors when I attempt to install clojure1.3, leiningen, and several other packages.   When I try the suggestion made in the error message, this is what happens:  $ sudo apt-get -f install  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following packages were automatically installed and are no longer required:   diffstat linux-headers-3.2.0-26-generic linux-headers-3.2.0-26 dh-apparmor dkms html2text libmail-sendmail-perl libsys-hostname-long-perl Use 'apt-get autoremove' to remove them. The following extra packages will be installed:   libjline-java Suggested packages:   libjline-java-doc The following NEW packages will be installed:   libjline-java 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. 23 not fully installed or removed. Need to get 0 B/72.0 kB of archives. After this operation, 129 kB of additional disk space will be used. Do you want to continue [Y/n]? Y (Reading database ... 226243 files and directories currently installed.) Unpacking libjline-java (from .../libjline-java_1.0-1_all.deb) ... dpkg: error processing /var/cache/apt/archives/libjline-java_1.0-1_all.deb (--unpack):  trying to overwrite '/usr/share/java/jline.jar', which is also in package scala 2.9.2-400 Errors were encountered while processing:  /var/cache/apt/archives/libjline-java_1.0-1_all.deb E: Sub-process /usr/bin/dpkg returned an error code (1)  $ sudo apt-get upgrade Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies:  bsh : Depends: libjline-java but it is not installed  groovy : Depends: libjline-java but it is not installed  rhino : Depends: libjline-java but it is not installed E: Unmet dependencies. Try using -f. How to get a DOM Element from a jQuery selector? I have a site on my webhotel I would like to run some scheduled tasks on. What methods of achieving this would you recommend?  What I’ve thought out so far is having a script included in the top of every page and then let this script check whether it’s time to run this job or not.  This is just a quick example of what I was thinking about:  if ($alreadyDone == 0 &amp;&amp; time() &gt; $timeToRunMaintainance) {    runTask();    $timeToRunMaintainance = time() + $interval; }    Anything else I should take into consideration or is there a better method than this? Why didn't (or did) I or some other user get a hat although the requirements were (or weren't) fulfilled? During Winter Bash, users can earn hats. But sometimes it seems that a hat was awarded even though its requirements weren't (yet) met, or vice versa.  What could be the reason for this? How can I animate the Restrict Visibility property of an object? I've been trying to create an animation where an object blinks on and off.   For example, I have a Cube and I set a keyframe on it's Restrict Visibility icon (the "eye" icon) with I, then I toggle off that "eye" so my cube doesn't show in the Viewport and again I press I to record that keyframe. Then, when I preview the animation in the Viewport, I see my cube appears and then disappears like I want, but when I render it, the cube does not disappear at all.  What am I doing wrong? Graph editor for shape key animation I have animation where object changes his shape using shape keys, and I want to make some animation smoothing with graph editor. Is there a way to use graph editor for shape key animation? What is a rocket engine thrusting against in space? I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity?