Let $m$ be a positive integer and let $\alpha$ be a real number. Show that $$\left[\alpha\right]+\left[\alpha+\frac{1}{m}\right]+\cdots + \left[\alpha+\frac{m-1}{m}\right] = \left[m\alpha\right]$$ where $\left[\ \ \right]$ is the integral part function.  Remark: Attempt to prove this result by induction but  I failed in the inductive step, I have been unable use the induction hypothesis.
I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:     DEFINITION Given  $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x &lt; z+1$$ and we denote it by $[x]$.   Now he asks to prove some basic things about it, such as: if $n\in \Bbb Z$, then $[x+n]=[x]+n$  So I proved it like this: Let $z=[x+n]$ and $z'=[x]$. Then we have that  $$z\leq x+n&lt;z+1$$  $$z'\leq x&lt;z'+1$$  Then $$z'+n\leq x+n&lt;z'+n+1$$  But since $z'$ is an integer, so is $z'+n$. Since $z$ is unique, it must be that $z'+n=z$.  However, this doesn't seem to get me anywhere to prove that  $$\left[ {2x} \right] = \left[ x \right] + \left[ {x + \frac{1}{2}} \right]$$  in and in general that   $$\left[ {nx} \right] = \sum\limits_{k = 0}^{n - 1} {\left[ {x + \frac{k}{n}} \right]} $$  Obviously one could do an informal proof thinking about "the carries", but that's not the idea, let alone how tedious it would be. Maybe there is some easier or clearer characterization of $[x]$ in terms of $x$ to work this out.  Another property is $$[-x]=\begin{cases}-[x]\text{ ; if }x\in \Bbb Z \cr-[x]-1 \text{ ; otherwise}\end{cases}$$  I argue: if $x\in\Bbb Z$, it is clear $[x]=x$. Then $-[x]=-x$, and $-[x]\in \Bbb Z$ so $[-[x]]=-[x]=[-x]$. For the other, I guess one could say:  $$n \leqslant x &lt; n + 1 \Rightarrow  - n - 1 &lt; x \leqslant -n$$  and since $x$ is not an integer, this should be the same as $$ - n - 1 \leqslant -x &lt; -n$$  $$ - n - 1 \leqslant -x &lt; (-n-1)+1$$  So $[-x]=-[x]-1$
While talking about ships and countries, is it a good practice to use the feminine form?  For example:  "Her economy" - while referring to a country's economy "Her flag (or deck etc)" - while referring to a ship  Is this practice common? Is it used today? I would like to open all chapters of my PDF ebooks in the same window but neither evince nor okular seems to be capable of doing that (at least not out-of-the-box).  Is there a way to work this out? How to I deal with unwrapped UVs orientation?   Delete the default Cube. Add a Cylinder Edit mode and edge select the Top and the Bottom and a single connecting edge. Make them into seams UV Unwrap (standard Unwrap mode)   The unwrapped UV now shows two circles above a row of quads. I can save this and open it in an image editor and paint over those areas and get the texture to align on the cylinder back in blender.  But, the pattern(just happens to be text) that was inserted in place of the row of quads is now upside down on the model. Any attempt to flip or align just makes things worse, from upside down to mirrored or the Top/Bottom not aligning.  So my real question is how could I have known that the quads represented an upside down view of the world?  Do other blender users export UVs and mark them with arrows and colours just to see how they align on a model before actually painting for real?  I tried selecting the edges in step 3 in different orders before making them into seams and it made no difference.  I also looked at the Normals but they appear to be only Inside/Outside not Up/Down.  Blender 2.70 Where can I ask a question about finding general software that meets certain requirements? I have a list of requirements for a program, but I don't know what program meets these criteria. That's about general-purpose software, for example, not a programming IDE or statistical workbench, questions about which would certainly go to respective sites.  Is this question appropriate on Stack Overflow? Where should I ask about it? On Super User or elsewhere? This may be a very basic question.      Let $R$ be a commutative ring with unity (may be assumed to be an integral domain, if necessary) and $I$ be an ideal of $R$. Assume that $R/I$ is isomorphic to $R$. Does this mean $I=(0)$ ? Changing trains in Lille - what's the transfer with luggage between Lille Europe and Lille Flanders stations like? Lille has two main train stations, Lille Europe and Lille Flanders. (Gare de Marchandises Saint-Saveur is no longer a station, and instead hosts art exhibitions and the like). Lille Europe is the new station on the high speed lines between London / Paris / Brussels, while Lille Flanders is the original station now used for a mixture of local and high speed trains.  For some rail journeys, changing trains in Lille is the best option (avoiding having to cross Paris to change trains), and for others it's the only option. Sometimes you get lucky and both trains use the same station in Lille, but not all.  For the cases when you find yourself needing to change between Lille Flanders and Lille Europe stations, what's the connection like? Does it take long? Is it outside / inside? Is it a faff with luggage, or is there a step-free route for most of it? Are you best to walk, or take public transport? And a lower concern, is there anything interesting to see of the town during the transfer? The difference between try/catch/throw and try/catch(e)/throw e when does a functor map products into products? I would like to add in QGIS real time weather data.  I found this  and this  and I tried to follow the example, but without any success.  I've added a WMS weather data layer for precipitation to the QGIS map, as follows:  First, the parameters for the service:  service = 'crs=EPSG:900913&amp;dpiMode=7&amp;featureCount=10&amp;format=image/png&amp;layers=precipitation&amp;styles=&amp;url=http://wms.openweathermap.org/service'   Next, create the raster layer, specifying WMS as the type:  rlayer = QgsRasterLayer(service, "precip", "wms")   Finally, add the precipitation layer to the map:  QgsMapLayerRegistry.instance().addMapLayers([rlayer])   the example didn't show any error in the code and no data was added to the weather layer either, I'm not sure why? Where are the review queue counters? The review queue counters have disappeared:        Where are they?  When I did screenshot the Close Votes queue was not empty, if someone think so. Is this series known to converge, and if so, what does it converge to (if known)?  Where $p_n$ is prime number $n$, and $p_1 = 2$,  $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$ Related (but not the same):         Earlier today I asked a question about aligning equations in an enumerate environment and got an answer that seemed to work for the example I gave, but it turns out I wasn't really specific enough for my problem.  I'm using align* environments, often as the only part of a list item. (If you're wondering why I'm not using more words to introduce the align*, these are mostly homework solutions where most of the work is algebra, etc.)  Here's my code and what it makes (above the middle line):  \begin{enumerate}   \item \begin{align*}     x^2 + y^2 &amp;= x^2 + (iy)^2 \\               &amp;= (x + iy) (x - iy)   \end{align*}   \item \begin{align*}     \frac{1}{n^2-4} &amp;= \frac14 \frac{4}{(n-2)(n+2)} \\                     &amp;= \frac14 \left( \frac{1}{n-2}-\frac{1}{n+2} \right)   \end{align*} \end{enumerate}     I'm trying to make the baseline of the first line of the align* environment the same as the number's baseline—I want the first line of the align* to have the same vertical position (centered horizontally in the same way) relative to its number as in the second half of my picture. (Side question: why is there so much space before the align* starts?) How do I backup my Stackexchange favorites for offline use? Teenage girl in life-sustaining valley on poisoned Earth visited by another survivor If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$? If I have a manifold $M$ and $\omega \in \Omega^q(M)$, considered an open subset $U$ of $M$, and choose a point $p \in M$, is it true that $(\omega|_{U})_p=0 \rightarrow \omega_p=0$? With $\omega|_{U}$ I mean the pull-back of $\omega$ by the embedding $F:U\to M$. The converse is obviously true, but this implication? My roommate put a hunk of beef roast in the slow-cooker for a good 6 hours.  It's a fairly large sized slow-cooker, at least 8 quarts.  The meat was cooked through but very chewy, and I never remember my mom having that issue with a slow-cooker.  My roommate tends to put only enough water to get halfway up the side of the roast, and always uses a slow-cooker 'bag' (another thing my mom never did).  What advice can I offer them to get a better outcome? How can I crack a 16 bit hash/CRC/checksum with a large sample set? How can I add an empty directory to a Git repository? How can I add an empty directory (that contains no files) to a Git repository? Here's my code:  x = 1.0 y = 100000.0     print x/y   My quotient displays as 1.00000e-05.  Is there any way to suppress scientific notation and make it display as 0.00001? I'm going to use the result as a string. What's a recommended alternative to Quickbooks pro? I've just installed Ubuntu to replace Windows. I can't seem to find a good alternative to  to run on Ubuntu. Any help would be appreciated.