How do I sum the series $$\sum_{n=1}^{\infty} \left( n\cdot \ln\frac{2n+1}{2n-1} - 1\right)$$  My attempts:   Then $nth$ term can be found to be $-\ln(3 \cdot 5 \cdots (2n-1)) + n\cdot \log(2n+1) -n$.    What to do after this?
how can I find sum of series $ \sum_{n=1}^{\infty} (n\cdot \ln \frac{2n+1}{2n-1} - 1) $? It is so weird for me because I put this to Mathematica and it tells me that sum does not converge...  Let consider sum no to infinity, but to n  $$ \sum_{k=1}^{n} (k\cdot \ln \frac{2k+1}{2k-1} - 1) =$$ $$ ln \frac{3}{1}\cdot \left(\frac{5}{3}\right)^2 \cdot...\cdot \left(\frac{2n+1}{2n-1}\right)^n - n = ln \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1} - n $$ but $$ n = ln e^n $$ so it will be $$ln\frac{1}{e^n} \cdot \frac{1}{1}\cdot \frac{1}{3}\cdot \frac{1}{5}\cdot ... \frac{1}{2n-1}$$  So the limit of it is $-\infty$ Have I done this well or I missed sth?
Question on the correlation between two dependent variables I'm working on this question and it's stumping me.     Let $S_n = X_1 + \ldots + X_n$ (with $n&gt;=1$) be a random walk with $X_1, \ldots, X_n$ be iid RV's.   $$ E(X_k)=\mu,\,{\rm Var}(X_k)=\sigma^2.  $$   Find the covariance of $S_n$ and $S_m$   Can anyone help out? I am trying to use the equation: $${\rm Cov}[S_n, S_m] = E[S_nS_m] - E[S_n]E[S_m]$$  but can't quite figure it out. How to wrap lines correctly inside algorithmic I am having problems displaying algorithms. The wrapping is really ugly and I can't, for the life of me, find a fix that doesn't involve manually tweaking the hell out of everything, to the point that I might as well scrap the crappy algorithm packages and write everything out in TeX.  Is there any way to properly indent the following  \documentclass{article} \usepackage{algorithm} \usepackage{algpseudocode} \begin{document} \begin{algorithm}    \caption{My pseudo code.} \begin{algorithmic}[1]    \While{this line is sooooooooooo long and boring and too much for algorithmic to handle}       \State look at this state, this state is just too long for algorithmic to display it properly        \While{again this line is sooooooooooo long and boring and too much for algorithmic too handle}          \State look at this state again, this state is just too long for algorithmic to handle, I'm just going to switch to Word       \EndWhile    \EndWhile \end{algorithmic} \end{algorithm} \end{document}​     so that the output is similar to Can a 20V 4.51A power supply be used with a computer that requires a 19V 3.42A power supply? Mask and inherit the same systemd unit Suppose a package provides whatever.service file so certain events (including whatever.socket) cause a daemon to start with a certain configuration.  I want to disable this completely. In particular, I want the old socket and any other unit files that might refer to this service by name to be unable to start the daemon.  Instead, I want to create my own unit files for this daemon with a different .socket unit and different environment variables. I also want my override to automatically take account of possible upstream updates to the original unit file (which my current solution of copying whatever.service as whatever-modified.service and editing it unfortunately does not provide).  If I create the following symlinks and files: /etc/systemd/system/whatever.service → /dev/null /etc/systemd/system/whatever-modified.service → /usr/lib/systemd/system/whatever.service /etc/systemd/system/whatever-modified.service.d/fix.conf  systemd considers the new service to be masked as well and refuses to start it. How to avoid this? With CSS, use "..." for overflowed block of multi-lines In a project the customer wants the images of the WMS to be secured against unauthorized access.   What are the common solutions to realize that?   Are they compatible with the OGC-standards? Loading a package conditionally Is there a clever way to show that  $$|x| - |y| \leq |x-y|$$  I believe I can think of a way to solve this using cases, but the book I'm working out of said that "A very short proof is possible if you write things in the right way" which has motivated me to figure out this quick way, but I've failed thus far and any help would be appreciated Are there LaTeX performance profiling tools? Why do quantum effects dominates when the thermal de Broglie wavelength is on the order of or larger than the interparticle distance? We say that when the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell–Boltzmann gas. On the other hand, when the thermal de Broglie wavelength is on the order of or larger than the interparticle distance, quantum effects will dominate and the gas must be treated as a Fermi gas or a Bose gas, depending on the nature of the gas particles.  $(V/N)^{1/3} &lt; \lambda_{th} $. My question is why do quantum effects dominate? The choice field adds a blank option in all my sharepoint lists. How do I remove this? the column settings does not have a blank option. Only half of the display works, can I set to display only in that region? I have a Macbook with a cracked screen, only the top left half of the screen works, is there anyway to reduce the screen resolution, keep the DPI the same, and thus have the entire OS visible in the top left section of the screen which works?  The below photo demonstrates what I mean, with parts of the creen totally unusable, it would make sense to be able to display the whole screen in the remaining usable part. Why am I getting a NoClassDefFoundError in Java? I am getting a NoClassDefFoundError when I run my Java application.  What is typically the cause of this? What happens if you plug more than one charger in the new MacBook Pro (2016)? Apple's  decision to convert all the ports on their 2016 MacBook Pro lineup to USB-C ports contributes to blur the difference between ports that supply power (such as a power adapters) and ports that can power external devices (such as USB keys, mice or other gadgets).    This made me think about the following scenario: you sit at a table where your laptop is plugged in on the right, and don't notice that it is plugged in, so you decide to charge it with a second nearby charger that is laying around (i.e. belonging to someone who lives with you and who recently sat at the same spot).  What would happen then?   Would the computer charge twice as fast, would one of the chargers be dismissed by the system (which one?), or would it generate a surcharge that would destroy the computer?  I couldn't find anything about this online, but since I am always trying to force a USB in the MagSafe charger of my MacBook Air, I think that I could easily end up making this kind of mistake.  Note: My question asks what happens when two chargers are inserted into a MacBook Pro, but I am also wondering what would happend if 3, or 4 chargers were inserted. Although this scenario is unlikely to happen by mistake, it would constitute an obvious vulnerability to an exposed device, i.e. in a public space. When trying to deploy the converted metadata using VS code sfdx force:mdapi:deploy deploy to my Production Org and I get the error Property 'itemInstances' not valid in version 48.0 and Property 'processOrder' not valid in version 48.0. But when I go to Developer Console and open a class, even though it says 48.0, I have the option to select 49.0, so I am assuming I have 49.0. Yet I get this error. Pretty much all my classes, trigger and components are version 48.0, but I am not sure how that affects flexipages and Approval processes. Blender render crash when using subsurf modifiers. Which are safe and unsafe regions in Iran? Now there's a lot of propaganda in the west that leads us to believe Iran is dangerous and travellers shouldn't go there.  Obviously there are major political disagreements between several governments and the Iranian government and this should be taken into account.  But at a human level I have met many Iranians travelling elsewhere in the world and have found them to be friendly, charming, and hospitable. I have also met several western travellers who spent time in Iran, including hitchhiking around the country, who only have wonderful things to say about the country and its people.  As a traveller I've learned to trust the most what I hear from other travellers about a destination, especially ones which prefer similar ways of travel to myself. I've also learned to trust the opinions of inhabitants of countries bordering on the destination country the least.  Yet some people still insist that Iran is not safe for a traveller, or that .  I can look up for myself the general travel advisories issued by governments and disclaimers in WikiVoyage. So please don't regurgitate those an answers.  I want to know from travellers who've been there, from Iranians, or from references to specific incidents targeting foreigners/westers about which parts of Iran are no problem, and which parts should be avoided, even by experienced intrepid travellers.  (I realize any advice won't apply to Israeli citizens unfortunately. I'm Australian.) How is acceleration measured if there is change only in the direction of velocity & not magnitude? The formal definition of acceleration is      It is the measure of how fast the velocity changes with respect to time.   Now, when the magnitude of velocity changes, acceleration $$\vec{a} = \frac{d \vec{v}}{dt}$$ and its unit is $m.s^{-2}$ .  But what about the case where the velocity only changes its direction keeping its magnitude constant? The definition can then be      Acceleration is the measure of how fast the velocity changes its direction with respect to time.   But how can I calculate the acceleration then?? What should be the unit then??? Please help. How to pad zeroes to a string? How do I blend multiple photos to remove random people from the background? I'm very excited about the new  for Android/iPhone. Let's say you're taking a picture of your friend in front of a landmark with several people walking by. The app allows you to remove the people from the photo, leaving only the background and your friend. The app isn't just doing a clone of the surrounding image to paint over unwanted subjects. It's actually taking multiple images, detecting which objects are moving, and letting you select them for removal. This sounds awesome! But I don't want to be limited to taking pictures with a smart phone.  I would like to take multiple photos of the same scene. (Maybe with a tripod, but it would be better if I didn't have to use a tripod.) Then I would somehow blend them together to remove unwanted elements. How do I do that? I know there are tons of photoshop tutorials out there, but all the ones I find just tell you how to use the clone tool.