How do I find this limit $ \lim \limits_{n \to \infty}(\dfrac{1}{n+1}+\dfrac{1}{n+2}.....\dfrac{1}{n+n})$  $(A)=\log_e1$  $(B)=\log_e2$  $(C)=\log_e3$  $(D)=\log_e4$  Please give me hint or something. What  i could see is $ \lim \limits_{n \to \infty}\dfrac{1}{n} \rightarrow 0 $ that is not even in the options.
For each positive integer $n$, let $x_n=1/(n+1)+1/(n+2)+\cdots+1/(2n)$. Prove that the sequence $(x_n)$ converges.   This is what I have so far..  Let $\epsilon &gt; 0$ be given to us. We must show that there exists an $N \in \mathbb{N}$  such that $n\ge N \implies \left|\frac{1}{2n} - 1\right| &lt; \epsilon$. Choose $N$ to be any positive integer which is larger than $\frac{1}{\epsilon}$. (So $N&gt;\frac{1}{\epsilon}$.) Then $n\ge N \implies \left|\frac{1}{2n} - 1\right| = |\frac{1-2n}{2n}|$
$\newcommand{\P}{\mathbb{P}}$Is it valid to interpret $\P(B|A)$ as "infinite probability" when $\P(A) = 0$? In the following sentence, I want to know if I've understand the correct meaning or not:     This version holds that the wave function does not so much describe the state of the physical system as the state of our knowledge of the system.    Can we say that it means the wave function describes the state of our knowledge  of the system more than it describes the state of the physical system  ?  It is worth noting that in the Cambridge Dictionary, we read:  not so much sth as sth ​ If you say that something is not so much one thing as something else, you mean it is more the second thing: They're not so much lovers as friends. I don't feel angry so much as sad. Let $f:\Bbb{R}\to\Bbb{R}$ be a function given by  $$f(x)=\begin{cases} \exp\left(\frac{1}{x^2-1}\right) &amp; \text{if }\vert x\vert\lt 1\\  0 &amp; \text{if }\vert x\vert\geqslant 1  \end{cases}$$  I would like to prove that $f\in C^\infty$, that is, $f\in C^k$ for all $k\in \mathbb{N}$. I think that it can be done by induction on $k$. If $\vert x\vert\gt1$, the problem is trivial. On other points, the base case is the simplest and the only that I'm be able to do. Can someone help me?  Thanks. Html Textarea elements only wrap when they reach a space or tab character. This is fine, until the user types a looooooooooooooooooooooong enough word. I'm looking for a way to strictly enforce line breaks (eg.: even if it results in "loooooooooooo \n ooooooooooong").    The best I've found is to add a zero-width unicode space after every letter, but this breaks copy and paste operations. Anyone know of a better way?  Note: I'm referring to the "textarea" element here (i.e.: the one that behaves similarly to a text input) - not just a plain old block of text. What is a natural response to &quot;Good to see you. How you doing?&quot;? Will a natural response include &quot;doing&quot; as in &quot;I'm doing okay&quot; or will &quot;I'm okay&quot; be more natural? How can I debug my JavaScript code? Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ Was Yoda wrong about the limits of telekinesis? In Empire Strikes Back, Yoda  that an object's size doesn't matter and that the difficulty of levitating an X-Wing was only in his mind. If the upper limits of telekinesis are determined by belief, what's to stop a Jedi/Sith from pulling a moon out of orbit, moving an entire solar system, or splitting an atom with the Force? Squeezing all the SEO out of a URL as possible I am working on an ecommerce site, I told our SEO consultant that I plan to make the URL scheme: /products/&lt;id&gt;/&lt;name&gt;. This is similar to Stackoverflow's URLs which are /questions/&lt;id&gt;/&lt;title&gt;.  He asked me if I could change the URL scheme to /p/&lt;id&gt;/&lt;name&gt; instead. I know why he wants this change, the word "products" isn't needed to find the correct product, and it doesn't offer any SEO, so shortening it to just p would make the relevant keywords in the &lt;name&gt; weigh more.  His main priority is maximizing SEO, but the part that I don't think he is considering is how this effects the semantics of the site. Also having the word "products" looks like it has meaning and a reason for being there, just having a p looks chaotic and ugly to me. I also don't think it makes that much of a difference does it? Stackoverflow doesn't use /q/&lt;id&gt;/&lt;title&gt; and they do just fine, I do realize that theres many factors at play here though, not just the URL.  So I want some outside opinions on which is the better way and why? Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$ Prove that for any natural number $n&gt;1$, we have $(2n)!&lt;(n(n+1))^n$    I tried using induction, but I failed in that approach, I rather found it was untrue, but on several case testing, I found it to be true. I am out of ideas here, please help. Thank you. How to fix the weird surface in fluid object Does the  tag position in HTML affects performance of the webpage? In this  to a question about a series, a theorem was stated:     If $A= \{a_i \}$ is a set such that $\sum_{i = 1}^{\infty} \frac{1}{a_i}$ converges, then $d(A) = 0$, where $d(A)$ is the  of the set.   My background in number theory is basically zero and all my attempts to prove this have been utterly unseccessful; would anyone please help me, or at least provide me with a hint to prove it? Accessing bash command line args $@ vs $* I'm currently working on the following problem for my computer theory class. It goes as follows:     Let $A$ and $B$ be regular expressions. Show then that $A^*B$ is the solution   of $X = AX + B$   Am I supposed to let $X = A^*B$? If so then   $A^*B = A(A^*B) + B$  I can understand that  $A(A^*B)$ could be reduced to $A^*B$ but what happens with the "${}+ B$" part of the expression? I've read up on Kleene algebra and found an axiom that seems relevant in which      $b + ax ≤ x → a^∗b ≤ x$   but I'm not actually sure if it applies. Is there a way to set app-specific volume? When are Opportunity Attacks provoked while holding a reach and a non-reach weapon? Assume I wield a Whip in one hand and a dagger in the other.  If a creature moves from 5ft to 10ft it has technically moved out of my reach (dagger), but at the same time it is still inside my reach (whip). When does it provoke opportunity attacks from me?  I am fully aware that this would not enable me to make off-hand attacks since the whip is not a light weapon, just to cut that issue out of the question. Proof that the largest eigenvalue of a stochastic matrix is $1$ top output: cpu usage > 100% Books for self-studying time series analysis?