Chemistry in Latex
Typesetting Chemical Formulas
Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$ I want to prove the following: $$\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$$ what I did is(use binominal): $$\binom{n}{k}X^k\cdot 1^{n-k} = (X+1)^n$$ $$k\binom{n}{k}X^k\cdot 1^{n-k} = k(X+1)^k-1$$ now I replace $k$ by $n$ and insert $X=1$ , I will get now $n\cdot2^{n-1}$ what you are suggesting? there is another way to do that? thanks! Is GNU's Java Compiler (GCJ) dead? Evaluating a sequence of complex numbers: $|z_n - z_m| < \frac1{1+ |n-m|}$ Let $\{z_n\}$ be sequence of complex numbers such that   $$ |z_n - z_m| &lt; \frac{1}{1+ |n-m|} $$ for all $n,m$  Given this information, can we compute $\lim_{n \to \infty} z_n $?  Attempt:  For sure the sequence is bounded since the RHS is always $\leq 1 $. IF $z_m = 0$, then we have   $$ |z_n | &lt; 1 $$ for all $n$. I claim that the limit is $1$. How can I prove this ? SyntaxError: Non-ASCII character '\xa3' in file when function returns '£' When should I use  or >()? I know that &lt;, &lt;(), &lt;&lt; all are used to give input and  &gt;, &gt;() which are used to redirect output.  But I don't know what is clear difference between them and when to use them.  Please explain in detail and along with references if possible. Thanks My HDRI doesn't appear on Cycles Render Finding an angle within an 80-80-20 isosceles triangle Search with Auto-Complete (like Gmail / Intellisense) finding examples for a non negative and continuous function for which the infinite integral is finite but the limit at infinity doesn't exist Question:  a. Find an example for a non-negative and continuous function s.t. $\int _0^\infty f(x)dx$ is finite but the following limit doesn't exist: $\lim_{x\to \infty} f(x)$.  b. Is it possible that $\int _0^\infty f(x)dx$ is bounded but $f(x)$ is not bounded?  What we did with A: We suggested the function $|sin(x^2)|$ which has periods that get smaller and smaller until they no longer imply on the sum. But since it's a trig function it doesn't have a determinate limit. Wolfram didn't have the integral value for that func, and we were wondering if it really converges and if it is really a good example.  What we did with B: We thought about the function: $f(x)=  { x\in \Bbb N },{e^{-x} \notin \Bbb N }$ and we had a disagreement whether f(x) is integrable. I said no because similarly to Dirichlet function, one sum's limit will be 0 while the other one's will be infinite. My partner said that if I find a $\delta&lt;1$ then the definition of Riemann's integral does hold and so this integral is equal to that of $f(x)=e^{-x}$ She was trying to say this functions integral will be 0, but still it won't be bounded. I disagreed. Who's right? Retagging questions in terms of possible answer technologies What is the $n$ times iterate of $f(x)=\frac{x}{\sqrt{1+x^2}}$? Difference between live and neutral wires In domestic electrical circuits, there are 3 wires - live, earth and neutral. What is the difference between the live and neutral wires?  As there is AC supply, it means that there are no fixed positive and negative terminals. Current rapidly switches direction. Does that not mean that both the live and neutral wires carry the same amount of current, alternatively? Why is the neutral wire called 'neutral'? Cannot find Background Images feature in Blender 2.8 Cannot find Background Images feature to add in a photo in Blender 2.8  Where is it? Understanding reference counting with Cocoa and Objective-C I'm just beginning to have a look at Objective-C and Cocoa with a view to playing with the iPhone SDK. I'm reasonably comfortable with C's malloc and free concept, but Cocoa's references counting scheme has me rather confused. I'm told it's very elegant once you understand it, but I'm just not over the hump yet.  How do release, retain and autorelease work and what are the conventions about their use?  (Or failing that, what did you read which helped you get it?) I ran into the problem that my primary key sequence is not in sync with my table rows.   That is, when I insert a new row I get a duplicate key error because the sequence implied in the serial datatype returns a number that already exists.  It seems to be caused by import/restores not maintaining the sequence properly. boolean[] vs. BitSet: Which is more efficient? What is more efficient in terms of memory and CPU usage — an array of booleans or a BitSet? Specific BitSet methods are not used, only get/set/clear (==, =, Arrays.fill respectively for an array). Word for the satisfaction of curiosity There is a good feeling you get when your curiosity is satisfied, especially if is about a question you found interesting or important, or have spent a lot of time and energy trying to answer. Depending on how much I wanted to know, this feeling can be quite pleasing. Is there a word specifically for either the satisfaction of a curiosity, or the feeling that results? Facebook message notification showing blank message Since I updated the system to 10.11.5, the Facebook message notification is not showing message text.   Is this a bug in this software upgrade of OSX? Callback functions in Java Let $p\in[1,\infty]$, $a\in\ell^\infty$ and $$T:\ell^p\to\ell^p\;,\;\;\;x\mapsto ax.$$ Note that $\left\|T\right\|=\left\|a\right\|_{\ell^\infty}$. It's easy to see that if $a\in\ell^1$, then $T$ is nuclear and hence compact.  Can we more generally conclude that $T$ is compact?  Assume $p&lt;\infty$. The intuitive approach would be to define $$T_n:\ell^p\to\ell^p\;,\;\;\;x\mapsto(a_1x_1,\ldots,a_nx_n,0,\ldots)$$ for $n\in\mathbb N$ and to show that $\left\|T_n-T\right\|\to0$. Now, $$\left\|(T_n-T)x\right\|^p=\sum_{i=n+1}^\infty|a_ix_i|^p\tag1$$ for all $x\in\ell^p$ and $n\in\mathbb N$. The right-hand side of $(1)$ clearly converges to $0$ as $n\to\infty$, but this only yields convergence in the strong operator topology. On the other hand, if $(a_n)_{n\in\mathbb N}$ would be nonincreasing and $a_n\to0$ (e.g. $a_n=\frac1n$), then $$\left\|(T_n-T)x\right\|^p\le|a_{n+1}|^p\left\|x\right\|_{\ell^p}^p\tag2$$ for all $x\in\ell^p$ and $n\in\mathbb N$ and we could immediately conclude $\left\|T_n-T\right\|\to0$. EDIT: Maybe the general result can be obtained from the convergence in strong operator topology (which holds by $(1)$) by noting that $\ell^p$ has the approximation property; but I'm not sure whether this is true.