I have to prove the serial representation of: \begin{equation*} \int^1_0 x^x\,{\rm d}x=\sum^{\infty}_{n=0}\frac{(-1)^{n-1}}{n^n}. \end{equation*}  It obtains: \begin{equation*} x^x=\sum^{\infty}_{n=0}\frac{(f(x))^n}{n!}. \end{equation*}  The function $f(x)$ is continuous in $[0,1]$.
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!
Performance difference between dot notation versus method call in Objective-C You can use a standard dot notation or a method call in Objective-C to access a property of an object in Objective-C.  myObject.property = YES;   or  [myObject setProperty:YES];   Is there a difference in performance (in terms of accessing the property)? Is it just a matter of preference in terms of coding style? My printer does not print light magenta. (Canon MP220 - CL-38) I have a Canon MP 220, with new cartridges. Prints fine everything, but it won't print light magenta. I cleaned it and deep cleaned it many times. I have no idea what could be the problem. Cartridge models: CL-38 and PG-37.  Any ideas ?  LE: Now it won't print magenta at all! Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$. I have a JavaScript object like the following:  var p = {     "p1": "value1",     "p2": "value2",     "p3": "value3" };   Now I want to loop through all p elements (p1, p2, p3...) And get their keys and values. How can I do that?  I can modify the JavaScript object if necessary. My ultimate goal is to loop through some key value pairs and if possible I want to avoid using eval. The question asks it all really. When referring to a master's degree, do you use an apostrophe or not? That is, is it "a master's" or "a masters"? Picklist dependent on 2 controlling picklist I have 3 picklist in an object Test1, Test2, Test3  1. Test2 is dependent on Test1  2. Test3 is dependent on both.. Test1 and Test2    I want to have Test3 picklist showing values accordindly. i.e  if U selected Test1 = First and Test2 = SA . I should be able to see "IST,ABC and PQR" in Test 3.   Is this possible??? Why should the substitution be injective when integrating by substitution? I made a silly mistake in evaluating some integral by using a non-injective $u$-substitution.  But why should $u$-substitutions be injective in the first place?  I reasoned in the following way: the formula $$ \int_{\phi(a)}^{\phi(b)}g(x)\ dx = \int_a^b g(\phi(t))\phi^\prime(t)\ dt $$ holds for a general $C^1$ function $\phi$, even if it is not injective.  When you calculate an integral of the form $\int_a^b f(\phi(t))\ dt$, to use the formula above from right to left, you should find a function $f$ such that $$ f(\phi(t)) = g(\phi(t))\phi^\prime(t), $$ which do not exist if $\phi$ is not injective, i.e., $\phi(t) = 0$ for some $t$. This is why substitutions should be injective.  Is my reasoning correct?  If so, I believe that if $\phi^\prime(t) = 0 \Rightarrow f(\phi(t)) = 0$, a function $g$ that satisfies the formula above may exist and $\phi$ should not necessarily be injective.  Is this right?  I am often confused about the fact $\phi$ should be injective.  Is there an intuitive way to interpret this fact, so that I always remember to take a $\phi$ that is injective?  I would be grateful if you could help me understand this matter. What are "schlieren" in English? The German word for these patterns is "Schliere, pl. Schlieren", and apparently this word has entered the English language as a loanword (cf. article  in Wikipedia or entry  in OED). But the German loanword is probably not something most native speakers of English understand without the help of a dictionary, so what would you call, or how would you describe, the following patterns?  Concrete:    Air:    Oil on water:    Coffee and milk:    Paint:    Smoke:    Rock:    Marbled paper:    And so on.    "Waves" is not applicable, because waves are vertical undulations of the surface, while these are a pattern on, or inside, the surface level, with no amplitude.  "Striæ" does not really fit either, because those refer more to the parallel layers, than to the curly forms. How do I set the page title? I know I could change the page title with drupal_set_title(t('Amy page title')), but when I try it in Drupal 8 I get the There is no such function error.  How can I change page title in Drupal 8? Find all solutions to $x^2\equiv 1\pmod {91},\ 91 = 7\cdot 13$ Can a ring of positive characteristic have infinite number of elements? I am just looking for some clarification on this exercise:  Let $A = \{a,b,c,d\}$. Give an example of a relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive.  I understand that if I was only dealing with $A$ and and not $A^2$ I could give an answer similar to:  $R = \{(a,a),(a,b),(a,d),(b,a),(b,b),(c,c),(d,a),(d,d)\}$  Here $R$ is reflexive, symmetric, and not transitive.  What is confusing me is the fact that we have $A^2$. I am not sure what I am missing here. I am looking through my book to see if I can figure this out, but if someone could help shed some light on this it would be appreciated.  Thanks,  Tony Newline in a WPF-label? How can I add a newline in the text of a label in WPF such as the following?  &lt;Label&gt;Lorem    ipsum&lt;/Label&gt; Trying to dual boot Ubuntu 14.10 with Windows 10, last one not detected I have a laptop, and I'm going to install Ubuntu (I have experience, and I have  successfully installed , Ubuntu, , , , , , etc.), but installing  (Utopic Unicorn) alongside , Ubuntu can't detect Windows.  It shows me like there's no OS, but I have Windows&nbsp;10.  I have two primary partitions by the way if I manually install. If the Ubuntu installer can't detect Windows&nbsp;10, will it be detected at ? MySQL Insert query doesn't work with WHERE clause What's wrong with this query: INSERT INTO Users( weight, desiredWeight ) VALUES ( 160, 145 ) WHERE id = 1;  It works without the WHERE clause. I've seemed to have forgot my SQL. I have just started special theory of relativity. The limiting speed known as speed of light fascinated me most. I asked my teacher:      Consider two massless objects moving in the same direction at the speed of light. What would be their relative velocity w.r.t each other?   My teacher tells me that their relative velocity would be zero. But then the speed of light is universal constant regardless of the motion of its frame of reference, so shouldn't their relative velocity be $c$? What is their relative velocity and how? Do objects moving at the speed of light obey law of addition of velocities? Seem to have lost 4 Silver badges and 10 Bronze badges How do we "restore" notepad after crash? I was running NotePad and Chrome when my computer suddenly shuts down.  After reboot, chrome was able to "restore" itself such that I didn't lose any data.  However, how do we "restore" notepad.exe? Frequency encoding is a widely used technique in Kaggle competitions, and many times proves to be a very reasonable way of dealing with categorical features with high cardinality. I really don't understand why it works.   Does it work in very specific cases where frequencies are correlated with the target or is it more general? What's the rationale behind it? Print file content without the first and last lines