I have the following sum to evaluate:  $$\sum\limits_{n=1}^{\infty}\frac{n}{2^{n+1}}$$ My professor says that I can evaluate this through "differentiating the geometric series" but I am not quite sure what that means. I already know that this converges to $1$, just not sure how to prove.
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
Is liquid cooling necessary for high CPU performance over extended periods? I'm planning on building a system for high CPU and memory performance.  Graphics performance is not a major concern, as I do not plan to use it for gaming.  I noticed that liquid cooling seems to have become common since my last build, and I am wondering if it is needed in my case.  I do not plan to overclock any of the components.  The intended use for the system is primarily development and occasional compute-intensive tasks that will run on all cores continuously for several days to a week.  The system will not normally be in continuous usage like this, but it needs to be capable of occasional extended periods of continuous use at full capacity.  I have included the main components below.  CPU: i7-5960X  Memory: 64GB DDR4  Graphics card: Nvidia GeForce GT 730 2GB  Drive 1: 1TB SSD  Drive 2: 4TB 7200RPM HDD  Power supply: 700W  Is liquid cooling needed for the above build?  If not, is there any advantage in using it?  I have read reviews that mentioned liquid cooling unit failures that resulted in leaks, so I would rather not use it if it is not necessary.  Thanks! Is it possible to check where an alias was defined? Is it okay to record audio for a lecture? Sometimes the Professor will say things quickly, or say things before I can write them down. It would be of great benefit if I could record audio for the course, but I am not sure the legal guidelines around doing so. I read in a CSV file for linking to a .shp file.   All the variables that are just numbers came in as text.   How do I change the text variables to numeric in QGIS? What is a good complex analysis textbook, barring Ahlfors's? I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.) Although elements like &lt;div&gt;s normally grow to fit their contents, using the float property can cause a startling problem for CSS newbies: If floated elements have non-floated parent elements, the parent will collapse. For example:   &lt;div&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    The parent div in this example will not expand to contain its floated children - it will appear to have height: 0. How do you solve this problem? I would like to create an exhaustive list of solutions here. If you're aware of cross-browser compatibility issues, please point them out. Solution 1 Float the parent.   &lt;div style="float: left;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Semantic code. Cons: You may not always want the parent floated. Even if you do, do you float the parents' parent, and so on? Must you float every ancestor element? Solution 2 Give the parent an explicit height.   &lt;div style="height: 300px;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Semantic code. Cons: Not flexible - if the content changes or the browser is resized, the layout will break. Solution 3 Append a &quot;spacer&quot; element inside the parent element, like this:   &lt;div&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt;   &lt;div class="spacer" style="clear: both;"&gt;&lt;/div&gt; &lt;/div&gt;    Pros: Straightforward to code. Cons: Not semantic; the spacer div exists only as a layout hack. Solution 4 Set parent to overflow: auto.   &lt;div style="overflow: auto;"&gt;   &lt;div style="float: left;"&gt;Div 1&lt;/div&gt;   &lt;div style="float: left;"&gt;Div 2&lt;/div&gt; &lt;/div&gt;    Pros: Doesn't require extra div. Cons: Seems like a hack - that's not the overflow property's stated purpose. Comments? Other suggestions? I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this:  Let $u=x^x$.  $$\begin{align} \int x^x \, dx&amp;=\int u \, du\\[6pt] &amp;=\frac{u^2}{2}\\[6pt] &amp;=\dfrac{\left(x^x\right)^2}{2}\\[6pt] &amp;=\frac{x^{2x}}{2} \end{align}$$  But it's certain that this isn't the correct way to evaluate that, and the answer must be wrong. Why is a ConcurrentModificationException thrown and how to debug it I am using a Collection (a HashMap used indirectly by the JPA, it so happens), but apparently randomly the code throws a ConcurrentModificationException. What is causing it and how do I fix this problem? By using some synchronization, perhaps?  Here is the full stack-trace:  Exception in thread "pool-1-thread-1" java.util.ConcurrentModificationException         at java.util.HashMap$HashIterator.nextEntry(Unknown Source)         at java.util.HashMap$ValueIterator.next(Unknown Source)         at org.hibernate.collection.AbstractPersistentCollection$IteratorProxy.next(AbstractPersistentCollection.java:555)         at org.hibernate.engine.Cascade.cascadeCollectionElements(Cascade.java:296)         at org.hibernate.engine.Cascade.cascadeCollection(Cascade.java:242)         at org.hibernate.engine.Cascade.cascadeAssociation(Cascade.java:219)         at org.hibernate.engine.Cascade.cascadeProperty(Cascade.java:169)         at org.hibernate.engine.Cascade.cascade(Cascade.java:130) Product of invertibles in $\Bbb Z_n$ [Wilson's Theorem generalization] How can we compute the product of all invertibles in $\Bbb Z_n$?  In the special case $n=p$ where $p$ is a prime it is Wilson's theorem. By pairing inverses it reduces to computing the product of all $a$ such that $a^2\equiv 1\pmod{n}.\,$  How can we do that? \nameref breaks for star versions of section/paragraph when titlesec is loaded \nameref breaks for the star versions of section, paragraph when titlesec is loaded. I have a document that numbers paragraphs by default, but I have a few I do not want to number but would like to refer to. These are marked with \paragraph*, however, if the titlesec package is loaded at the same time, it causes nameref to display an empty string (the link box of hyperref is still visible).  MWE:  \documentclass[12pt]{article} \usepackage[sf]{titlesec} %comment this out and it works fine. \usepackage{hyperref} \begin{document} \section{Sections} \label{sec} This is \nameref{sec} \paragraph{A named Paragraph} \label{par} This is \nameref{par} \paragraph*{A starred named Paragraph} \label{par:star} This is \nameref{par:star} \end{document}   I suspect titlesec causes this defect for ALL star variants of sectioning commands but have not tested this. For me only paragraph* is relevant however a fix for all variants would be the best.  I have a hunch it might be related to this:  however this happens to go way beyond my latex knowledge and is just "magic" for me. Correct tracked camera orientation I have been playing around with camera tracking and I'm happy enough with the tracking and alignment of the markers with the footage when viewed through the camera.    The problem is, when I am not looking through the camera I can see that the grid isn't aligned to the camera.    How do you correctly align the grid to this tracked camera? I have set a floor and two points for scale, also clicked setup tracking scene, but this just added cube and plane which were on the grid. I have a Lenovo Ideapad with pre-installed Windows 8.1 In order to install Ubuntu I disabled secure boot and changed boot mode from UEFI to Legacy. This allowed me to install Ubuntu and the grub loader is also showing an option to either boot into Windows or Ubuntu, but whenever I try booting to Windows it says Cannot load.  This happens because the boot order is set to Legacy mode and Windows 8 will boot only with UEFI. I am unable to access my BIOS from where I can change the boot order. I press F1/F2 on booting but it doesn't load the BIOS, and directly goes to the grub loader screen.  How can I make my Windows 8 run along with Ubuntu? What is the intuition behind defining completeness in a statistic as being impossible to form an unbiased estimator of $0$ from it? Difference between == and === in JavaScript How do I send a file as an email attachment using Linux command line? Network interface name changes after update to 15.10 - udev changes I updated a virtual machine running Ubuntu server 15.04 to the just released new stable 15.10.  There were no apparent problems, but after reboot the machine lost its network connection. lspci showed the correct card and lsmod proved that the driver had been loaded.  Hints about editing or removing /etc/udev/rules.d/70-persistent-net.rules did not succeed (still no network, no new file), the famous /lib/udev/write_net_rules did not even exist.  I don't have much clue regarding udev and systemd, so it took me some time to find in /sys/class/net that the network interface name changed. It became ens32, which was renamed from eth0. After I changed /etc/network/interfaces network communication was enabled again.  But - why does this happen at all? And if it happens, shouldn't the interfaces file be updated automatically?  Now that I deleted the virtual network card and created another one, eth0 becomes eno16777728, which is not so, uhm, nice.  Should this behaviour (at least that of the upgrade procedure) be reported as a bug?  Furthermore, is /etc/udev/rules.d/70-persistent-net.rules obsolete now? And (although only partly related) what about /etc/udev/rules.d/70-persistent-cd.rules?)  Edit - additional question (after accepted answer) The dependency of /etc/network/interfaces from the actual interface name bothered me. I tried a new installation of *ubuntu in another virtual machine. To my surprise, /etc/network/interfaces contained a reference to the loopback device only. However, with my original VM this change prevented the network to come up. So what else do I need to change?  Edit Well, from another test I found that a new Ubuntu Server has the network interface card in /etc/network/interfaces, while XUbuntu has not. Maybe it is using network-manager or something else. Why isn't persistence working on Lubuntu 12.04 Live-USB? How can I display site logo, name and slogan? Distribution of a ratio of uniforms: What is wrong? Suppose that  $X$ and $Y$ are two i.i.d. uniform random variables on the interval $[0,1]$  Let $Z=X/Y$, I am finding the cdf of $Z$, i.e. $ \Pr(Z\leq z) $.  Now, I came up with two ways of doing this. One produces a correct answer consistent with the pdf here: , the other does not. Why is the second method wrong?  First Method  $\newcommand{\rd}{\mathrm{d}} \Pr(Z\leq z) = \Pr(X/Y\leq z) = \Pr(X\leq zY) = \int^{1}_{0}\int^{\min(1,zy)}_{0} \rd x \rd y = \int^{1}_{0}\min(1,zy)\ \rd y$ $  = \left\{      \begin{array}{lr}        \int^{1/z}_{0}zy\ \rd y + \int^{1}_{1/z} \rd y&amp; : z &gt; 1\\        \int^{1}_{0}zy\ \rd y &amp; : z \leq 1      \end{array}    \right. $ $  = \left\{      \begin{array}{lr}        1 - \frac{1}{2z} &amp; : z &gt; 1\\        \frac{z}{2} &amp; : z \leq 1      \end{array}    \right. $  This appears correct.  Second Method  $\Pr(X/Y\leq z) = \Pr(X \leq zY\ |\ zY \geq 1)\Pr(zY \geq 1) + \Pr(X \leq zY\ |\ zY &lt; 1)\Pr(zY &lt; 1) $ by total probability  $ = \Pr(X \leq zY\ |\ zY \geq 1)\Pr(Y \geq 1/z) + \Pr(X \leq zY\ |\ zY &lt; 1)\Pr(Y &lt; 1/z)$  Taking $z&gt;1$ yields $ (1)(1-\frac{1}{z}) + (\int^{1/z}_{0}\int^{zy}_{0} \rd x \rd y)(\frac{1}{z}) = 1-\frac{1}{z} + (\int^{1/z}_{0}zy\ \rd y)(\frac{1}{z}) = 1-\frac{1}{z} + \frac{1}{2z^{2}}$  This is already different. Why is this wrong?  Thanks! Based on this question: , LVS may be the right solution for my problem.   But when I went to its homepage , I found that LVS has been updated since Nov, 2008. The world's moving fast, and I don't know if LVS was obsolete or not.  Is LVS standing still, or there're some better solutions to replace it?  Thank you so much.