Show that $$\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}&gt;1,\:\forall n\in\mathbb{N}$$ This is a 9th grade problem. I was trying to take the greatest numerator, which is the  last numerator of the last fraction. But there are only $2n+1$ terms. Right? After that I have no idea. Thx!
Prove by induction: $$\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} &gt; 1$$  adding $1/(3m+4)$ as the next $m+1$ value proves pretty fruitless. Can I make some simplifications in the inequality that because the $m$ step is true by the inductive hypothesis, the 1 is already less than all those values?
Why does my wife's skin buzz when she's using her laptop? When my wife uses her laptop, if I touch her skin, I can feel a buzz.  She doesn't feel the buzz, but she can hear it if I touch her ear.  So I'm guessing it's a faulty laptop, and she's conducting an electrical current.  But why would she not feel anything, and what would it be that she would be hearing when I touch her ear?    More info:  The effect is only intermitent - it's pretty reliable in a single session on the laptop, but some sessions it won't happen and others it will.  I had the same sensation with a desk lamp that I had several years ago, with no moving parts (as far as I could tell)  The effect only occurs when I move my finger - if I'm stationary, I don't notice anything.   I was playing with my son, and noticed the same buzz.  First I thought he was touching the laptop.  Then I realised he had skin-to-skin contact with my wife who was using the laptop. Review Count - Fix It or Lose It I had booked a ticket for travel through Indian Railways using IRCTC and got a confirmed ticket. While booking, I forgot to mention the details of a one year old child who will also be traveling.     Will it create any problem while boarding the train? Captured variable in a loop in C# Let $ \mathbb R _ { &gt; 0 } $ be the set of positive real numbers. Find all functions $ f : \mathbb R _ { &gt; 0 } \to \mathbb R _ { &gt; 0 } $ such that $$ f \big ( x y + f ( x ) \big) = f \big( f ( x ) f ( y ) \big) + x $$ for all positive real numbers $ x $ and $ y $.  What I thought: We could change $ x $ by $ y $, and then subtract. Source: Brazil National Olympiad 2019 #3 How to choose the default gcc and g++ version? I've tried to find any solution or proof for $$\langle x| x' \rangle=\delta(x-x'),$$ but I only came to this post:   So I got the information, that the vector $|x\rangle$ form a dirac-normalized basis for the Hilbert Space.   I know that the dirac-delta distribution is defined like this: $$\delta(x-x') = \begin{cases} 0 &amp;\mbox{if } x\neq x' \\ \infty &amp; \mbox{if } x=x' \end{cases},$$ this means that my x' is a point on my x axis where I have my infinite high peak. And also $$\int_{-\infty}^{\infty}dx\cdot \delta(x-x')=1.$$  But how actually correlate this with the scalar product of vectors x, x' in the Hilbert Space that form a so-called 'diracl-normailzed" basis of it?  Can you give me some tips on this please? Or maybe you actually know a link, where this is explained. How to dump a man page? Don't count non-printing characters, HTML comments or other content not displayed toward minimum character requirements How to remove a vector QgsVectorLayer from QGIS python plugin I managed to add QgsVectorLayer in QGIS through a python plugin, for example:  vl = self.iface.addVectorLayer(uri.uri(), layerName, self.dbConn.getProviderName())   But how can I remove it? I've placed a command block making it day all the time. I want to turn it off, but I've lost it. What's the quickest way to find it? Is it same number of candies for any stage of evolution while transferring pokemon to professor? I was just wondering if the number of candies we get while transferring the pokemon to professor is same for any stage of evolution ?   For example, you would get a pidgey candy for transferring a pidgey. Is it same number of candies for transferring pidgey, pidgeotto or pidgeot ? Converse to Chinese Remainder Theorem So as seen on this question , we know that if $(n,m) \neq 1$, then $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, because the right hand side does not have an element of order $nm$.  But take a more general setting. Let $R$ be a commutative ring, and let $A,B$ be ideals in $R$. $A$ and $B$ are said to be comaximal if $A+B=R$. If $A, B$ are comaximal, then we have: $R/AB \cong R/A \times R/B$. In this setup, is the converse true?      If we have $A, B$ ideals in $R$ such that $R/AB \cong R/A \times R/B$, do we always have that $A$ and $B$ are comaximal? Some ships in Star Trek are referred to with definite articles. "The Enterprise", "The Defiant". Other ships are referred to without them. "Voyager", "Enterprise". Is there a pattern to or explanation for this behavior? I am trying cmd --> MMC --> Server Manager and then connecting to my Server 2012 no GUI  but i keep getting the following error The WinRM client cannot process the request  I am unable to upload images right now  This is not related to other post "How to Remote Manage Windows Server 2012 in Server Manager?" because title and description of that post is quite different from what i am asking Check if email exist php Is the intersection of a closed set and a compact set always compact? I am going through Rudin's Principles of Mathematical Analysis in preparation for the masters exam, and I am seeking clarification on a corollary.  Theorem 2.34 states that compact sets in metric spaces are closed. Theorem 2.35 states that closed subsets of compact spaces are compact. As a corollary, Rudin then states that if $L$ is closed and $K$ is compact, then their intersection $L \cap K$ is compact, citing 2.34 and 2.24(b) (intersections of closed sets are closed) to argue that $L \cap K$ is closed, and then using 2.35 to show that $L \cap K$ is compact as a closed subset of a compact set.  Am I correct in believing that this corollary holds for metric spaces, and not in general topological spaces? If $n^c\in\mathbb N$ for every $n\in\mathbb N$, then $c$ is a non-negative integer? Supposing that a real number $c$ is given, is the following true?  "If $n^c$ is a natural number for every natural number $n$, then $c$ is a non-negative integer."  Though this seems true, I can't prove that. Can anyone help? Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$ Calculate $\tan9^{\circ}-\tan27^{\circ}-\tan63^{\circ}+\tan81^{\circ}$?  The correct answer should be 4. Context: I want to store the TIFF data to use it for running a linear regression model later on. Ultimate goal: Create a carbon map using field data as a basis. Libraries: library(sp) library(raster) library(tidyverse) library(rgdal)  plotcoord &lt;- SpatialPointsDataFrame(AGCplot[,3:4], AGCplot[,1:2],proj4string=CRS('+proj=longlat'))  pspcoord&lt;-spTransform(plotcoord, CRS('+proj=utm +zone=21')) #Omzetten naar UTM21N (= S1) plot(pspcoord)  s1 &lt;- stack('Some.tif')  colnamen = c(&quot;plotid&quot;,&quot;AGC&quot;,&quot;B2&quot;,&quot;B3&quot;,&quot;B4&quot;,&quot;B5&quot;,&quot;B6&quot;,&quot;B7&quot;,&quot;B8&quot;,&quot;B8A&quot;,&quot;B11&quot;,&quot;B12&quot;,&quot;NDVI&quot;,&quot;NDWI&quot;,&quot;MMRI&quot;,&quot;IRECI&quot;,&quot;MASI&quot;,&quot;VV&quot;,&quot;VH&quot;)  s1_e &lt;- extract(s1,pspcoord,method='bilinear')  Upon extraction I get the following message: Warning message: In proj4string(x) : CRS object has comment, which is lost in output