Why is $\lim_{n \to \infty} \int_{0}^{1} { {nx^{n-1}} \over {1+x} }dx=1/2$? The function sequence $f_n=nx^{n-1}$ tends to zero for all $x \in [0,1)$. Why is the above limit not equal to $0$? Can you show it with Lebesgue's convergence theorems?
Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$ Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$.   \begin{align}   s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx  \end{align}  I have written a solution of my own, but I would like to know it is completely correct, and also I would like if people post more alternative solutions.
Iterate enum values using java generics $g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$ If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left&lt;g\right&gt;\oplus H$  Attempt: Using fundamental theorem I know that $G=C_{n_1}\times\cdots\times C_{n_k}$ where $n_i|n_{i+1}$. With some work I proved $|\left&lt;g\right&gt;|=n_k$, so $C_{n_k}\cong \left&lt;g\right&gt;$. Then $G\cong H\times \left&lt;g\right&gt;$, but this is not an equality. What does first authorship really mean? In my field (theoretical computer science), authors of any paper are always listed alphabetically; our papers .  (Well... hardly ever.)  In most other disciplines, at least within science and engineering, the ordering of authors is a signal about their relative contributions to the paper, with the first author indicating the most significant contributor.  Hiring and promotion committees do give extra weight to "first-author papers" (and sometimes have to be reminded that not all areas have them).  As an outsider, I find this practice confusing.  What does first authorship actually mean in your discipline? I understand vaguely that the first author is supposed to be the one who "", but what counts as "work" in this comparison?  Does "most" mean "more than all the other coauthors together" or just "more than any other coauthor"?  What happens when the comparison is unclear?   How often is "did the most work" the actual truth, versus a cover story for a more complex political decision?  I realize that the precise answer is different for every paper.  I'm looking for general guidelines for how an outsider (like me) should interpret first authorship in your field.  Pointers to guidelines from journals or professional societies would be especially helpful.  Please give only one answer per discipline. Provisioning Profiles menu item missing from Xcode 5 How can I show that $\frac{1}{1+|z|^2}$ isn't analytic at $z=1$? How does Mage Armor interact with Unarmored Defense? How can a capacitive touch screen be triggered without human contact? I want to robotically touch an iPhone screen without any human intervention.   I've done some experiments but have not found a reliable solution. It seems even styluses that work through gloves still rely on the capacitive characteristics of the human body.    Carrot held by human works Carrot held in plastic clamp doesn't Pogo Stylus works held by human and held by human through clothes Stylus does not work held in plastic clamp.  Stylus works held in plastic clamp attached to human through clothes via jump lead. Stylus held in plastic clamp with jump lead dangling off seems to work most the time.   Is it possible to mimic the human element using capacitors and other components? What signal does the capacitive screen need to be triggered?  Thanks Live refresh shows "modified" on tagged questions even when unmodified Where can I find the location of folders for installed programs? no space left in '/' but a lot space left in '/home' i install ubuntu and followed a guide to put 20gb in '/' and the rest (150gb) in '/home'. Now i just found out that a lot of the things installed in ubuntu usually goes to '/' instead of '/home'. I have a lot more things to install.  So what is the point of '/home' ? How do I fix this ? Should I transfer 150gb from '/home' to / ?  Filesystem      Size  Used Avail Use% Mounted on udev            7.8G     0  7.8G   0% /dev tmpfs           1.6G  9.5M  1.6G   1% /run /dev/sda6        19G   16G  1.8G  91% / tmpfs           7.8G  676K  7.8G   1% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           7.8G     0  7.8G   0% /sys/fs/cgroup /dev/sda2        96M   29M   68M  31% /boot/efi /dev/sda8       159G  1.1G  150G   1% /home tmpfs           1.6G   56K  1.6G   1% /run/user/1000   what is /boot ? Its not '/' or '/home' Type safety: Unchecked cast In my spring application context file, I have something like:  &lt;util:map id="someMap" map-class="java.util.HashMap" key-type="java.lang.String" value-type="java.lang.String"&gt;     &lt;entry key="some_key" value="some value" /&gt;     &lt;entry key="some_key_2" value="some value" /&gt;    &lt;/util:map&gt;   In java class, the implementation looks like:  private Map&lt;String, String&gt; someMap = new HashMap&lt;String, String&gt;(); someMap = (HashMap&lt;String, String&gt;)getApplicationContext().getBean("someMap");   In Eclipse, I see a warning that says:  Type safety: Unchecked cast from Object to HashMap  What did I do wrong? How do I resolve the issue? Why is filling a balloon from your mouth much harder initially? Why is it that when you first fill up a balloon, it's hard to get air through, but after inflating it a bit, it becomes much easier to further inflate the balloon? My application got crashed because of not initializing the view Since upgrading to 13.10, I can't log in to unity desktop. Light dm works correctly, but attempting to log in tries to start the session then drops back to light. I've already dropped to terminal (ctrl+alt+f2) and done this:  sudo apt-get update  sudo apt-get install --reinstall ubuntu-desktop  sudo apt-get install unity   Logging in as a guest session also fails. Logging in to other window managers works with varying degrees of success.  Note: I have Cinnamon 2.0 installed from PPA.  I'm using a 2 monitor setup.  Also of note is that the session prior to my upgrade to 13.10 the background of unity failed to display at all, instead showing what was there in the screen buffer from the previous frame. The entire OS worked correctly otherwise though, so I just ignored it for the session. No other upgrades or even updates were done prior to this occurring.  My upgrade path to 13.10 was basically this: Install 13.04 alongside Windows 7, use ubuntu as a glorified web browser for a while, get updates (in preparation for 13.10), install 13.10.  I also used Unity Tweak Tool to change some aspects of unity, particularly auto-hide.  Any help or ideas would be appreciated, as I'm typing this on my phone :( Is storing passwords hashed with the email more secure than without? Almost all standard sources on the existence of Majorana spinors (e.g. Appendix B.1 to Polchinski's "String Theory", Vol. 2) do so in a way I consider inherently ugly:   A priori, we are dealing with an irreducible complex representation $(V,\rho)$ of the Clifford algebra of signature $(p,q)$, i.e. generalized Dirac spinors. That Majorana spinors exist means abstractly that there is a  on $V$, i.e. a conjugate-linear map $\phi : V\to V$ with $\phi^2 = \mathrm{id}_V$ that commutes at least with the $\mathfrak{so}(p,q)$ action.  Every single source I can find for Majorana spinors uses operations like the transpose, complex conjugation and Hermitian adjoint on the $\Gamma$-matrices to obtain matrices acting on the same space. This is abstractly wrong, the transpose acts on the dual, the complex conjugation on the conjugate, and the Hermitian adjoint needs an inner product we have no reason for choosing. Of course, since $V$ is finite-dimensional, one can pick a basis and define the uncanonical isomorphisms to its dual and its conjugate, but I find this inelegant, particularly since the standard derivations require us to make a particular such choice with respect to the signs the $\Gamma$-matrices have under e.g. ${}^\dagger$. Finally, the Majorana spinors are usually defined by some equation involving an unnatural and arbitrary-looking product of $\Gamma$-matrices, which varies from source to source according to different sign conventions and sign choices made in the course of the derivation.  It's inelegant because the rest of the theory of spinors can be developed without making such uncanonical choices. Both the uniqueness of the dimension of the irreducible Dirac representations (there are two of them in odd dimensions) and the existence of the Weyl spinors in even dimensions can be derived purely from the abstract properties of the Clifford algebra, no choices made, no transpose, adjoint or conjugate occuring. The (admittedly slightly subjective question) is: Is there a way to show in which dimensions Majorana spinors exist that neither requires an uncanonical choice of basis nor arbitrary choices of signs?  Some partial results:   In even dimensions, the Dirac representation is necessarily self-conjugate since it is the only irreducible representation of the Clifford algebra, so all that is left to show is that a conjugate-linear $\mathfrak{so}$-equivariant map on it squares to $\mathrm{id}_V$ and not to $-\mathrm{id}_V$. However, I can't seem to exhibit any particular equivariant map on it that one could simply check for its square. In odd dimensions, one first needs to figure out whether the two inequivalent Dirac representations are conjugate to each other or self-conjugate.     As further motivation that a clear proof using only canonical properties of the Clifford algebra itself is required, consider the confusing and contradictory claims in the literature:   Polchinski, "String Theory", Vol. 2, p.434: $\mathrm{SO}(d-1,1)$ has Majoranas for $d = 0,1,2,3,4 \mod 8$, corresponding to a special case of $p-q = d-1-1 = 6,7,0,1,2 \mod 8$. Fecko, "Differential Geometry and Lie Groups for Physicists", pp. 651: $\mathrm{Cliff}(p,q)$ has Majoranas for $p-q = 0,2\mod 8$.  This clearly conflicts with Polchinski's claims e.g. for $d=3$. Figueroa-O'Farrill, , pp. 18: We have Majoranas for $p-q = 0,6,7\mod 8$ and "symplectic Majoranas" for $p-q = 2,3,4\mod 8$.   Note that these results conflict simply by the number of possible $p-q$ regardless of whether I've correctly taken care of the differing conventions of whether $p$ or $q$ denotes timelike dimensions. biblatex and new line for DOI, URL and Eprint Interior of closure of open set in topological space If $U$ is an open set in a topological space, is it true that $U$ is the interior of the closure of itself?   If $U$ is open, it must be the interior of itself. But is it the interior of the closure of itself? In the closure, we include all points $x$ such that any open set containing $x$ also contains a point in $U$. In the interior of the closure, we take the union of all open sets contained in the closure. Limit as $x\to 2$ of $\frac{\cos(\frac \pi x)}{x-2} $ I need an adjustment layer which will invert in the same way as when you select some text in a text layer