Show that $ \overline \lim\limits_{n \rightarrow \infty}\left(\frac{1+a_{n+1}}{a_{n}}\right)^{n} \geq \mathrm{e} $
An inequality from Littlewood's Miscellany
Fitting linear model / ANOVA by group What happens if a Rainbow Servant takes levels in Mystic Theurge? Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$ I am looking for the combinatorial argument for the identity:  \begin{equation} k\binom{n}{k} = n\binom{n-1}{k-1} \end{equation}  This is easy to show algebraically as:  \begin{equation} \binom{n}{k} = \dfrac{n(n-1)(n-2)(n-k+1)}{k(k-1)!} \end{equation}   What is the combinatorial argument? What are some general ideas to get started?   Here is a clarification of 2. From what I have seen so far, proving (combinatorially) an identity with an addition sign usually implies that we need to partition a set (this makes sense because of the addition rule and provides a nice visual). On the contrary, the previous observation leads me to believe that multiplication in identities can be resolved with the multiplication principle, but what is the "visual/interpretation" for this? Could someone provide such an interpretation for the example identity given above? Environment variables when run with 'sudo' How to find unused/dead code in java projects What tools do you use to find unused/dead code in large java projects? Our product has been in development for some years, and it is getting very hard to manually detect code that is no longer in use. We do however try to delete as much unused code as possible.  Suggestions for general strategies/techniques (other than specific tools) are also appreciated.  Edit: Note that we already use code coverage tools (Clover, IntelliJ), but these are of little help. Dead code still has unit tests, and shows up as covered. I guess an ideal tool would identify clusters of code which have very little other code depending on it, allowing for docues manual inspection. How does a spinning object "know" that it is spinning? I am constructing a thought experiment about a spinning object that is floating in intergalactic space.  I assume that this object is about the size of a planet so that it will have enough gravity so that a Foucault pendulum will work, although I'm not sure that this is necessary for the thought experiment.    I can easily determine that this object is spinning if I stand on the object and observe the galaxies around me rise and set.  Similarly, I can determine the axis of rotation.    For simplicity, I go to one of the poles of my object, and I set up my Foucault pendulum.  What will I see?  And why will I see it?  I assume that in intergalactic space, the gravity is very small (galaxies are very far away, and the mass of my object is very small compared to the mass of a galaxy) so that there will be little coupling between the gravitational field of my object and the gravitational field of the galaxies around me. How to make password-less login work I generated a key and copied it to my target:  ssh-keygen ssh-copy-id username@hostname   I left the the location of the key as default and did not set any passphrase.  For a small while after that, I was able to ssh username@hostname without being prompted for a password, and then it stopped working for some reason. generic function with a "has property X" constraint? Both sides differentiation expression with integral I have a problem with derivation of some function. This is related to computing the Poincare Map of Logistic Population model with Periodic harvesting. This function is:  $\phi(t,x_0) = x_0 + \int_{0}^{t} f(s,\phi(s,x_0)) ds $,  where $\phi(t,x_0)$ is a function: $\mathbb{R}  \times \mathbb{R} \rightarrow \mathbb{R}$.  I'm trying to differentiate  equation of this function both sides  with respect to $x_{0}$. In book from this equation comes from the result of this differentiation is:  $\frac{\partial \phi}{\partial x_0} (t,x_0) = 1 + \int_0^{t} \frac{\partial f}{\partial x_0} (s, \phi(s,x_0)) \cdot \frac{\partial \phi}{\partial x_0} (s, x_0)ds$.  I can't understand expression under the integral. In this book it is explained that it is clear from chain rule.Could someone explain to me this transformation in easy way? I will be grateful for your help Best regards A novel where a woman from "future" of 1999 speaks leet-speak If a series converges, then the sequence of terms converges to $0$. Stack Overflow reputation in CV Should I mention my Stack Overflow (and other Stack Exchange sites) reputation in my CV while applying for a post-graduate position? Show reference label when it is undefined How to combine paths in Java? Is there a Java equivalent for  in C#/.NET? Or any code to accomplish this?  This static method combines one or more strings into a path. I often have to sort a dictionary, consisting of keys &amp; values, by value. For example, I have a hash of words and respective frequencies, that I want to order by frequency.  There is a SortedList which is good for a single value (say frequency), that I want to map it back to the word.   orders by key, not value. Some resort to a , but is there a cleaner way? As mentioned on :     For something that can be done so   easily by accident, there should be   the ability to undo mistakes.   I know that I have clicked several times on a comment up-vote without actually wanting to.  I would suggest being able to cancel the up-vote (maybe only for a couple of minutes or even less than a minute, no problem, as long as I can undo a miss-click). When you pick up items in Skyrim it tells you the value of each item. However when I go and sell the items the merchants are only paying 30% of this value, at best.  Discounting any perks or other effects, how high does Speechcraft have to be before merchants start paying full value for items? A square with side $a$ is given. What is the average distance between two uniformly-distributed random points inside the square?  For more general "rectangle" case, see . The proof found there is fairly complex, and I am looking for a simpler proof for this special case. I expect it could be significantly simpler.  See also . Why should the minimal class be avoided? Term for things like "naughty step" where the step is not what is naughty Can anyone remind me of the grammatical term for the apparent misapplication of an attributive adjective, as in the phrase "the naughty step" (where it is not the step itself that is naughty but the person consigned to it)?