Proving the following statement
Series as an integral (sophomore's dream)
I have a VIP skybag (I believe this does not matter). It has a 3-number TSA lock built into the suitcase. I know the combination to unlock, let's say it's 345. Recently it was locked and while there was a different number on the combination lock the reset/set lock button was pressed.  Since then the reset/set lock button is pressed (stuck) and the lock button also does not unlock. Any pointers on how to unlock it? I am thinking of unscrewing the lock from inside of the suitcase but it seems quite a hassle, as I may have to assemble it back. Let me know if there is any other option to unlock it from the outside. Using Vim I often want to replace a block of code with a block that I just yanked.  But when I delete the block of code that is to be replaced, that block itself goes into the register which erases the block I just yanked. So I've got in the habit of yanking, then inserting, then deleting what I didn't want, but with large blocks of code this gets messy trying to keep the inserted block and the block to delete separate.  So what is the slickest and quickest way to replace text in Vim?   is there a way to delete text without putting it into the register? is there a way to say e.g. "replace next word" or "replace up to next paragraph" or is the best way to somehow use the multi-register feature? Not enough free disk space when upgrading Iterate a list as pair (current, next) in Python Is "connected, simply connected" Redundant? Here are my definitions of &quot;connected&quot; and &quot;simply connected.&quot;  A topological space $X$ is connected if and only if it is not the union of two nonempty disjoint open sets. A topological space $X$ is simply connected if and only if it is path-connected and has trivial fundamental group (i.e. $\pi_1(X)\simeq\{\mathrm{e}\}$ and $|\pi_0(X)|=1$).  It is a classic and elementary exercise in topology to show that, if a space is path-connected, then it is connected. Thus, if a space is simply connected, then it is connected. Yet, despite this implication, I've read several cases where the words &quot;connected, simply connected&quot; appear together. For example, in Kobayashi and Nomizu's Foundations of Differential Geometry, Volume 1, page 252, the following is written: &quot;Let $M$ be a connected, simply connected analytic manifold with an analytic linear connection.&quot; In  and  (Links to Journal Storage, also known as JSTOR), they use &quot;connected, simply connected&quot; on their first page. The first of these takes it a step further, referring to &quot;closed, compact, connected, simply connected $4$-manifolds,&quot; where a closed manifold (unfortunate terminology) is defined as a compact manifold without boundary. My question is simple: Why use the words &quot;connected, simply connected&quot; when being simply connected implies connectedness? Thank you. Game states & passing data between scenes Data inconsistency prohibition if a table refers to another via two many-to-many relationships I have following database design (by =&gt; foreign key constraints are depicted):   Company[id] CompanyRealm[id, company_id=>Company.id]many2many  Project[id, company_id=>Company.id] ProjectRealm[id, project_id=>Project.id, company_realm_id=>CompanyRealm.id]many2many   Problem is that provided database design allows inconsistent data. For example:   Company1(id=1) Company2(id=2) CompanyRealm(id=11, company_id=1) Project(id=33, company_id=2) ProjectRealm(id=44, project_id=33, company_realm_id=11)   (Company can have many Realms, Project belong to Company and  may concern any number of company's Realms)  ProjectRealm refers to two different companies:   Company1 (via CompanyRealm); and Company2 (via Project).    Is something wrong with my DB design? If yes - which prescriptions are violated? If no - how to prevent incorrect data insertion (via constraint? trigger with exception?) lsusb relevant output Bus 001 Device 006: ID 0bda:8812 Realtek Semiconductor Corp. RTL8812AU 802.11a/b/g/n/ac WLAN Adapter  Upon plugging the device it is recognized by lsusb, however when I go to wireless settings no adapter is recognized. The driver is installed (by going to software and updates -> addition drivers and selecting the open source driver). I'm getting errors in a matrix equation. When I compile it looks fine, but I'd like to fix them so I don't have to press return for every error at compile time. The code for the equation is:  \begin{equation*}  \begin{bmatrix} U(1,t)  &amp; \cdots &amp; U(n,t) \\  \vdots &amp; \ddots &amp; \vdots \\ O_{n1} &amp; \cdots &amp; O_{nn}  \end{bmatrix} \times \begin{bmatrix} W(1,1)  &amp; \cdots &amp; W(1,n) \\  \vdots &amp; \ddots &amp; \vdots \\  W(n,1) &amp; \cdots &amp; W(n,n)  \end{bmatrix} = \begin{bmatrix}  D(1,t)  &amp; \cdots &amp; D(n,t) \\ \vdots &amp; \ddots &amp; \vdots \\ I_{n1} &amp; \cdots &amp; I_{nn} \end{bmatrix}  \end{equation*}   The errors I'm getting are related to missing $'s inserted. How to restrict awk math output to 2 decimal places My Command is below and its print Free CPU and Used CPU below format.  I want to restrict awk output to 2 decimal place.  grep 'cpu ' /proc/stat | awk '{usage=($2+$4)*100/($2+$4+$5)} {free=($5)*100/($2+$3+$4+$5)} END {print "Used Cpu:",usage "%"} {print "Free Cpu:",free "%"}'   Output :   Free Cpu: 98.9588%  Used Cpu: 1.04125%   Desired output :  Free Cpu: 98.95%  Used Cpu: 1.04% Is Java "pass-by-reference" or "pass-by-value"? According to the well ordering theorem "". Whenever we have a well ordering on a set, it is not difficult to construct a new well ordering with a largest element. My question is:     For a given infinite set, can we find a well ordering on it such that there is no largest element? I have a question about the corollary to theorem 5.12 in Rudin's Principles of Mathematical Analysis (page 108):  Suppose $f$ is a real differentiable function on $[a,b]$ and suppose $f'(a)&lt; \lambda &lt; f'(b)$ then there is a point $x \in (a,b)$ such that $f'(x) = \lambda$ Corollary: If $f$ is differentiable on $[a,b]$ then $f'$ cannot have any simple discontinuities on $[a,b]$.  Can someone help me to show how he uses the result in the &quot;main theorem&quot; in the corollary?  (There are two cases of simple discontinuities  $f(x+) = f(x-) \neq f(x)$ and $f(x +) \neq f(x-)$ What would be an English equivalent for the Mexican Spanish word tocayo? Where do I get the key for the first house? I've 'completed' the game, but didn't find the key for the first house. (I got the one for the arcade machine house). Where can I find it? Setting up PyCharm to reference ArcPy? How would I setup  to recognize import arcpy?   I'd like to use PyCharm to develop and debug geoprocessing tool. Why normal acceleration doesn't bring a change in speed? Suppose There's a particle of mass m moving with speed $v_{0}$, at certain moment a force starts acting on it (centripetal force) and it start uniform circular motion  Now every other reference states that centripetal force only brings about change only in direction  But plugging-in things in the equations paint a different picture  suppose  normal/centripetal acceleration = $A$  then $|v| = \sqrt{v_{0}^2+\frac{(A t)^2}{1}}$  we can certainly see that factor of $t$ will bring change in magnitude of velocity   Secondly, why the velocity is always tangential to the circle, but how can we mathematically show such? Intuitively for me if there's constant acceleration toward a certain then at some point of time, the body should 'slack' and fall toward the direction of acceleration   Here' a picture for a reference: EJB 3.1 Dependency Injection Failed Definite Integral $\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$ $$\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$$  This is easy to evaluate with complex analysis but is there an elementary way (substitution, partial fractions, integration by parts)? In Dummit and Foote, Exercise 15.18 (p. 727 3rd edition) one is asked to prove that for a commutative ring with unity $R$, and $f$ a non-nilpotent element of $R$, that the localization $R_f$ is isomorphic to $R[x]/(xf - 1)$.  I have done this, using the strategy outlined in the problem, using the universal property of $R_f$ proved in Theorem 36. The problem I have is this:  If $R_f \cong R[x]/(xf - 1)$ and $\phi:R[x] \to R_f$ is a surjective homomorphism, it stands to reason that $\text{ker}(\phi)$ should be $(xf - 1)$. I cannot show this directly. The surjective homomorphism I have in mind is, for:  $p(x) = a_0 + a_1x + \cdots + a_mx^m$  $\phi(p) = \dfrac{a_0}{1} + \dfrac{a_1}{f} + \cdots + \dfrac{a_m}{f^m}$  It is clear to me that $\phi$ annihilates $(xf - 1)$, so that ideal is contained in the kernel. What I cannot show is the other containment, or equivalently:  If $\phi(g) = 0_{R_f}$ then $g(x) \in (xf - 1)$. The trouble I am running into is that $R$ might be a "bad" ring, containing a lot of zero-divisors, $f$ might even be a zero-divisor (being non-nilpotent is a considerably weaker property), so the usual polynomial tricks of the division algorithm or using the irreducibility of $xf - 1$ aren't much use, here.  I feel I am missing something obvious....what is it?