Using a definite integral find the value of $\lim_{n\rightarrow \infty }(\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n})$
The limit of truncated sums of harmonic series, $\lim\limits_{k\to\infty}\sum_{n=k+1}^{2k}{\frac{1}{n}}$
Git keeps prompting me for a password $\gcd(a,b)\!=\!1\!=\!\gcd(a,c)\Rightarrow\gcd(a,bc)\!=\!1$ [coprimes to $\,a\,$ are product closed] How do I go about proving this?  If $\gcd(a,b)=1$ and $\gcd(a,c)=1$, then $\gcd(a,bc)=1$. I'm very confused with gcd proofs. I've lost my portal home, and I am invincible. How can I leave the nether? Totally disconnected space that is not $T_2$ How do I make ditto marks? In France, students are taught not to repeat the same text twice when there is little difference between a line and the one below. It's a quick way to take notes.  I reproduced this effect by tinkering:    The code is as follows:  \documentclass[aspectratio=1610]{beamer}  \usepackage[T1]{fontenc} \usepackage[french]{babel}  \usepackage{amsmath, amsfonts, amssymb} \usefonttheme[onlymath]{serif}  \begin{document} \begin{frame} Lorsque le \og \textit{carreau carré} \fg{}   mesure: \begin{itemize} \item $\rm 1\,cm$ de côté, l'unité d'aire est appelée le \og \textrm{centimètre carré} \fg{} et est noté $\rm cm^2$; \item $\rm 1\,dm$ \hspace{1mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}''\hspace{8mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{décimètre carré} \fg{} \hspace{2mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1.5mm} $\rm dm^2$; \item $\rm 1\,m$ \hspace{3mm}'' \hspace{3mm}'' \hspace{8mm}'' \hspace{8mm}'' \hspace{5mm}'' \hspace{7mm}'' \hspace{5.5mm}''\hspace{1mm} \og \textrm{mètre carré} \fg{}  \hspace{9mm}'' \hspace{2mm}'' \hspace{4mm}'' \hspace{1mm} $\rm m^2$; \item etc. \end{itemize} \end{frame} \end{document}   How can I get the same effect more simply than by tinkering like I just did? How to choose the number of hidden layers and nodes in a feedforward neural network? Is there a standard and accepted method for selecting the number of layers, and the number of nodes in each layer, in a feed-forward neural network? I'm interested in automated ways of building neural networks. Number of non-decreasing functions? Let $A = \{1,2,3,\dots,10\}$ and $B = \{1,2,3,\dots,20\}$. Find the number of non-decreasing functions from $A$ to $B$.  What  I tried: Number of non-decreasing functions = (Total functions) - (Number of decreasing functions) Total functions are $20^{10}$. And I think there are ${20 \choose 10}$ decreasing functions. Since you choose any $10$ codomain numbers and there's just one way for them to be arranged so that the resultant is a decreasing function. However my answer doesn't match. Where am I going wrong? How can I directly compute the non-decreasing functions like without subtracting from total? What's the difference between "Service" and "/etc/init.d/"? What items are mutually exclusive between worlds? As of 1.2, some features are mutually exclusive between worlds.  In the case of ores at least, .  However, a  gives you the chance for a  (and the upgraded Sweetheart Necklace) which does not seem to have an equivalent version in the Corruption.    What items, if any, are truly mutually exclusive in function between worlds? How can I export a list containing e-mail addresses for everyone that has sent me an e-mail?  I have all of the messages archived. I sometimes have long running processes that I want to kick off before going home, so I create a SSH session to the server to start the process, but then I want to close my laptop and go home and later, after dinner, I want to check on the process that I started before leaving work. How can I do that with SSH? My understanding is that if you break your SSH connection you will also break your login session on the server, therefore killing the long running process. Changing ChildOf constraint influence to 0, but keeping the position it had as a child I had a sword object rigged with a single bone, with a ChildOf constraint making it a child of a sheath (with an influence of 1). Throughout an animation i move the sheath, and the sword bone follows as expected.  However, i cant seem to lower the influence of the sheath to zero and still have the sword retain the position it was just at, and I am wondering how i should go about making this happen.  Thanks for your time. Is there any formula for number of divisors of $a \times b$? Let $a$ and $b$ be two numbers,    Number of divisors of $a$ is $n_1$; Number of divisors of $b$ is $n_2$;    How to find the number of divisors $N$ of product $a \times b$ by using known number of divisors of factors $n_1$ and $n_2$?  If $a = 20$ and $b = 21$, then $n_1 = 6$ (number of divisors of $a$), $n_2 = 4$ (number of divisors of $b$. Then $a \times b = 420$ and $N = 32$ (number of divisors of $a \times b$).  Is there a formula to find $N$ in terms of $n_1$ and $n_2$? TAR 6 GB worth of files into separate 1.9 GB TAR files Title says it all, I don't see confirmation in  and a cursory google search seemed to say assume no and speculate about the pokemon bank. A twice differentiable function $f$ satisfies $f′′(x)+f(x)=−xg(x)f′(x)$, $\forall x\ge 0$. Consider $f\in C^2$ so that $$f''(x)+f(x)=-x\,g(x)f'(x), \ \forall x\ge0 $$ where $g(x)\ge 0$. Then ($\forall x\ge 0$)         (A) $f(x)^2+f'(x)^2$ is non-increasing,   (B) $f(x)^2&lt;3f(0)^2+(2f'(0))^2$,   (C) $|f(x)|\le\alpha$, where $\alpha$ is a fixed real constant.   (D) $\lim_{x\to\infty} f(x)\sin\left(\dfrac{1}{x}\right)$ exists.        Answer Given: (A),(B),(C),(D)  What I have tried : I differentiated option (A) and then with the help of the given equation in the question I was able to show that  it is less than zero . Hence I was able to infer that option (A) is correct .   Now coming to option (B).  I assumed the function in option (A) to be $h(x)$.   $$ h(x) = f(x)^2 + f'(x)^2 $$  Since $h'(x)&lt;0$ ,therefore $h(x)$ is a decreasing function   hence for all $x\ge 0$  $h(x)&lt; h(0)$  which implies $(f(x))^2 + (f'(x))^2&lt;(f(0))^2 + (f'(0))^2$  From the above expression we can get the second option .   Is my method correct ?. If not , please show the right method .  I am unable to get the the 3rd and 4th option . Given  $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$,  and $\mu_X \approx 0$, looking for:   accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$ asymptotic (?) approximation of same product    This is a special case $\mu_X \approx 0$ of a more . How to add a dot after the section number? I wonder is there any way to change the format of section numbering from 1.1 to 1.1. ? I know this is trivial, but I really need to add one more dot after the section number. Does anybody know how to do that?  Thanks a lot To my understanding, "Control" can have two meanings in statistics.   Control group: In an experiment, no treatment is given to the member of the control group.  Ex: Placebo vs Drug: You give drugs to one group and not to the other (control),  which is also referred as "controlled experiment".  Control for a variable: Technique of separating out the effect of a particular independent variable. Some of the other names given to this techniques are, " accounting for", "holding constant", "controlling for", some variable.  For example: In a Football viewing study (like or not like),  you may want to take out the effect of gender as we think gender causes bias, that is, male may like it more than female.     So, my question is for point (2). Two questions:  How do you "control"/ "account for" variables, in general. What techniques are used? (In terms of regression, ANOVA framework).   In above example, does choosing male and female randomly constitutes control? That is, is "randomness" one of the techniques for controlling other effects? How can there be animals on the surface in Zion's world? It is apparently canon that  in Neo's day.  However, in the first movie, The Matrix, Morpheus tells Neo that one of the things known for certain is that humans "burned the sky" in order to deny solar engery to the machines, which in turn led to the humans who survived the War being plugged into the Matrix to provide bioelectric energy to supply the machines.  If there's not enough sunlight for solar power, how can there be enough to grow plants to serve as food for animals?  Low levels I can understand -- house plants, at least, thrive in conditions too dim for solar panels to produce cost-effective power -- but every surface scene we see in the trilogy is as dark as a cloudy night; light coming only from the machines.  That wouldn't begin to support photosynthesis, which means all animal life would starve in, at most, a matter of a few years.