I need help with this exercise.  I need to prove $$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$  I think I should use some convergence theorem, but I'm stuck.  Thanks a lot!
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.  One series that popped into our discussion was $\sum\limits_{n=1}^{\infty} \frac{1}{n^n}$.  Proving the convergence of this series is trivial but finding the value to which converges has defied me so far. Mathematica says this series converges to $\approx 1.29129$.  I tried Googling about this series and found very little information about this series (which is actually surprising since the series looks cool enough to arise in some context).  We were joking that it should have something to do with $\pi,e,\phi,\gamma$ or at the least it must be a transcendental number :-).  My questions are:   What does this series converge to? Does this series arise in any context and are there interesting trivia to be known about this series?   I am actually slightly puzzled that I have not been able to find much about this series on the Internet. (At least my Google search did not yield any interesting results).
How do I run executable scripts in Nautilus? Change date format [YYYYMMDD] to [MM/DD/YYYY] I generated a report that formats date as 20160509 and my Google Sheets locale is already set to United States. I've played around with various options within Format > Number > Date Format but I can't change the dates to MM/DD/YYYY format without doing a simple find and replace. How do I accomplish this? Why are integer literals with leading zeroes interpreted strangely? This prints 83  System.out.println(0123)   However this prints 123  System.out.println(123)   Why does it work that way? Why is a linear transformation called linear? Consider the following file:  \documentclass{beamer}  \setbeamercolor{title}{bg=red} \setbeamercolor{author}{bg=yellow} \setbeamercolor{date}{bg=} \setbeamercolor{institute}{fg=red,bg=green}  \title{TitelTitelTitel} \author{meiermueller} \institute{Institute} \date{lalala}  \begin{document}  \usebackgroundtemplate{\color{blue}\rule{20cm}{15cm}}    \begin{frame}    \titlepage  \end{frame} \end{document}   On one of my computers (using MiKTeX 2.9 and beamer 2015/01/05 3.36), the result is the following:    Note how, in particular, the date box has a transparent background  On my other computer, I installed a new MiKTeX version (2.9.6800), including beamer 2018/09/02 v3.52, and get this:    Here, the date field has a white background, and I cannot seem to get it to be transparent.   Apparently, the difference has some relation to the beamer version, or maybe some other change in the new MiKTeX version.  But what ist the reason, and how do I get transparent backgrounds? Is this a plant or a fungus? It was soft and covered in flies. My guess is fungi.  Location: South Florida. I'm using a /tellraw command to display some kind of menu to the player. How can I react to the clickEvent without posting something in the chat? When I try using the scoreboard, I get an output in my chat like this Set score of &lt;objective&gt; for player &lt;playername&gt; to &lt;value&gt;. The same thing happens, when trying /setblock, /entitydata or /trigger Is there any way, I can execute a command which commandblocks can detect without writing a success/fail message in the chat?  the only commands without a "real" response message I found were /say, /tell and /tellraw. But all of these are impossible to detect(also, they write something else in the chat). What do (lambda) function closures capture? Why can't variables be declared in a switch statement? I've always wondered this - why can't you declare variables after a case label in a switch statement?  In C++ you can declare variables pretty much anywhere (and declaring them close to first use is obviously a good thing) but the following still won't work:  switch (val)   {   case VAL:     // This won't work   int newVal = 42;     break; case ANOTHER_VAL:     ...   break; }     The above gives me the following error (MSC):     initialization of 'newVal' is skipped by 'case' label   This seems to be a limitation in other languages too.  Why is this such a problem? Can I accept an already rejected edit (to my own post)? Today,  to one of . Before I got to see the notification, the edit already was rejected by other people (three rejections, one approval).  But the edit was actually a good one (like most edit proposals to my post I get to see) – a correction of syntax errors, so I would have definitively approved it.  (The suggested edit didn't change the main point of the answer, which was about the API to use, but it made the code example of my answer actually usable. I didn't compile it when writing my answer, it was late at the evening.)  So I did edit my post to include the proposed change, so the proposal is not fully lost. But the proposer only got the negative feedback from the rejection, and none of the reputation bonus an accepted edit gives.  My suggestion: Could there be added a way a post author can accept edits which were already rejected by other users?  Also, is there some way I can contact the rejectors and teach them not to reject good edits? I understand that in a lot of big-lab fields it is common for the principal investigator to append their name to a paper even if they did not write the paper, design the experiment, or collect data since they spend energy securing funding, and managing the whole lab. What about for small labs?  What are the requirements for a supervisor to be included as an author on a paper, as opposed to just appearing in the acknowledgements? If you are working on your own projects independently of your supervisor, but using funding provided by your supervisor (how does this change when the funding provides resources versus just your salary), are you suppose to add them as authors or just acknowledge the source of funding? No internet connection after ubuntu server 20.04 install, ifconfig not available I recently installed Ubuntu Server 20.04 LTS on my desktop, and found out there was no internet connection even though I had the LAN cable connected. ifconfig and iwconfig failed saying the corresponding packages net-tools and wireless-tools were not installed.   EDIT  Here is the info requested. Please find the image Is it safe to use a voltage divider for Arduino and DC motor controller? Opposite of "straight talk" What is the opposite for the straight talk idiom? How do I best call the activity when someone makes a very long preamble before he says what he wants? Get all non-unique values (i.e.: duplicate/more than one occurrence) in an array Short sci-fi story about Earth blockaded by other human colonies Like the subject: short sci-fi story about Earth blockaded by other human colonies.  The best humans have colonized some stellar systems. But they begin to treat humans of earth like inferiors. A war began, colonies won the war and blockaded the earth, but despite this the earth won. Who is the author? Does mass distribution/center of mass affect free fall? An everyday is example would be cars. If you were to drop a car, not taking air resistance into account, would a front-heavy car tumble forward as it fell? And a car with a perfect 50/50 front/rear weight distribution fall straight? Does mass distribution/center of mass affect free fall? Or would they both fall straight down without tumbling? Can you explain why please? Difference between DispatchQueue.main.async and DispatchQueue.main.sync Using zypper, how can I remove all packages installed from a specific repository? Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$ I need to show that $$\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$$  I know that $\displaystyle \binom{n}{k}\binom{k}{i}$ is counting the number of ways to pick $k$ elements from $n$ and then $i$ elements from those $k$, so $\displaystyle \sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i}$ is counting the number of ways to do that $\forall k$.  I know that $\displaystyle \binom{n}{i} = \binom{n}{n-i}$ and that $2^{n-i}$ is the number of elements in the power set for $[n]\setminus[i]$.  How do I show that both sides are equivalent? I'm stumped on this one. Could anyone give me a pointer in the right direction?