For $a_1, ... , a_2 &gt; 0$,  I need to prove that $(  \sum_{i=1}^{n} a_i)( \sum_{i=1}^{n} \frac{1}{a_i})\geq n^2 $.  When do we have inequality?  Should I do it by induction? any hints
$\sum_{i=1}^n(a_i)\sum_{i=1}^n(\frac{1}{a_i})\ge n^2$ is what I want to prove, where $a_i&gt;0$ and I need to showwhen equality holds.  I have tried using Cauchy-Schwarz, where inside the sums I multiply each value by 1 to do it but it doesn't work out.  I have also done  $\sum_{i=1}^n(a_i)\sum_{i=1}^n(\frac{1}{a_i})=\sum_{i=1}^n \sum_{i=1}^n(\frac{a_i}{a_i})=\sum_{i=1}^n\sum_{i=1}^n(1)=\sum_{i=1}^n(n)=n^2$, but this proves an equality.  Am I not allowed to do the first step I did in my work?
How to mount EXT4 disk on 10.15? Since I have osxfuse installed it would be nice if I could use it. There are a  (heavily) outdated  here, saying that it is not . Is this still the case? Can the present perfect continuous construct be used in passive voice? Can the passive voice be used for sentences in the present perfect continuous?  I understand the present perfect and its passive voice (have/has built, has been built). However, the passive voice of present perfect continuous is unclear:   Active: They have been building this house for years. Passive: ???   I have seen suggestions of “This house has been being built for years,” but it sounds awkward and I don't recall anywhere else where two be verbs follow one another. Is this the right form to use? piping in real time This is more a conceptual thing than a current problem.  Would it be possible to do this:  tail -f &lt;file&gt; | grep "string" | sed 's/stuff//g' &gt;&gt; output.txt  The problem I'm thinking is that tail -f never terminates so it won't progress in the command. I'm trying to read and output a file in real-time. If this doesn't work how would I do it? I'm a bit of a force.com IDE newbie - still learning. I'm trying to migrate a simple custom object (no fields) from one sandbox to another and I'm running into this error:  # Deploy Results:    File Name:    package.xml    Full Name:  package.xml    Action:  NO ACTION    Result:  FAILED    Problem: Cannot deploy InstalledPackage in Package Manifest with any other types!   Any help would be greatly appreciated.  Thanks! Reentering serial number in Windows 7 Does anyone know how to install a new serial on a working copy of Windows 7 - I get notified that my software is not genuine but cannot reenter the serial Server socket program not executing after connection establishment SQL Server nvarchar(max) vs nvarchar(n) affects performance The Sharpshooter feat includes an interesting feature: you can choose to take -5 penalty to hit with a ranged weapon attack to increase its damage by 10. Normally, I'd dismiss the trade-off as too situational, but I've only just thought of combining it with the Archery fighting style (+2 to hit with ranged weapon attacks).  After building a sample character using these features, the end result looks quite effective. My questions are: Is this build valid? Have I interpreted the rules correctly? If so, how well does it compare against other damage-focused builds?  5th level variant Human Fighter:   Ability Scores (via point buy): Str 10, Dex 16, Con 16, Int 9, Wis 12, Cha 10 HP: 47, AC: 17 (or 18 with disadvantage on Stealth) Weapons: Hand Crossbow x 2 Fighting Style: Archery Martial Archetype: Battle Master (with the Precision Attack maneuver) Feat (variant Human): Crossbow Expert Feat (4th level ability score improvement): Sharpshooter   The end result is a fighter capable of hitting targets 120' away, behind 1/2 or 3/4 cover, or engaged in melee with him, all without penalty. Normally, he can make 3 such attacks per round (Attack action + Extra Attack + Crossbow Expert's bonus attack). Once per short rest he can make 5 attacks per round via Action Surge (Attack action 1 + Extra Attack, Attack action 2 + Extra Attack, Crossbow Expert's bonus attack).  His normal attack bonus is +8 (3 proficiency, 3 Dexterity, 2 Archery fighting style), and normal damage is 1d6+3. Not amazing damage, but nice accuracy and flexibility (attacks can be focused on one target, or spread around; works equally well in melee and ranged combat).  The real winner is his ability to take -5 on any of these attack rolls to increase the damage by 10, to an average of 16.5 per shot (excluding crits). If he lands his usual 3 attacks, that's ~47 damage; 5 attacks is ~83. While the -5 to hit seems imposing, he's still got superiority dice: he can increase his attack roll by 1d8 after seeing the result, up to 4 times per short rest. And even without the superiority dice, his attack bonus is still an acceptable +3; ok for hitting low AC targets, or those granting advantage on attack rolls (e.g. surprised or prone + adjacent).  With a decent AC, a Fighter's HP, and the Second Wind feature, he can also take a decent amount of damage. He'll also have decent Strength, Dexterity, and Constitution saving throws.  This build seems almost too well rounded. Unlike a blaster, he only requires short rests to recover his big hitters. Unlike a brawler, he can target just about anything on the field at a moment's notice. Have I missed any rules, or overlooked any drawbacks? Best method for short time-series I have a question related to modeling short time-series. It is not a question , but how. What method would you recommend for modeling (very) short time-series (say of length $T \leq 20$)? By "best" I mean here the most robust one, that is the least prone to errors due the fact of limited numbers of observations. With short series single observations could influence the forecast, so the method should provide a cautious estimate of errors and possible variability connected to the forecast. I am generally interested in univariate time-series but it would be also interesting to know about other methods. Get full path of the files in PowerShell Lightning Change Event Is there a way to catch the change event of a particular field/component? I saw on the AuraDocs that there is a valueChange event but trying to handle it makes the app throw an error on load. I was wondering if I could build some automatic field validation / search suggestions without relying on other js libraries. Say I'm flying from Sydney, to Los Angeles (S2LA), back to Sydney (LA2S).  During S2LA, travelling with the rotation of the earth, would the flight time be longer than LA2S on account of Los Angeles turning/moving away from our position?  Or, in the opposite direction, would the flight to Sydney be faster since the Earth turns underneath us and moves Sydney closer?  ===   Please ignore jet stream effects and all other variables; this is a control case in an ideal environment. By "dramatically" I suppose I mean a delay of 1 hour or more. I was able to install Ubuntu 12.04 using the alternative version of the image, (64 bit) but when I start the system it tells me the GPU  locked up, and instead presents a text only screen.  I understand that this site is not for bug reports, but at the same time, I know that there are some versions of the installer than have to be compatible with my graphics card (Nvidia GTS 360M).  I assumed that the mere fact that the alternative installer ran was an indication that I could get some sort of graphical interface running after the install.  Is there some way of getting some type of graphical interface running on my machine? Even better would be is there a way to get the rest of the system running as normal (i.e. GPU up and running)?  If an internet connection is required to download driver support or something similar, directions for how to configure a wireless connection with WPA2-Personal protection from the text-only interface (bash).  Any help would be appreciated.  Note: I was able to find  which tells me that the card is supported.  Note 2: In the install software stage of the install, I was unable to install anything (no internet connection)  Update: The exact error message is [8.591394] [drm] 0000:01:00.0: GPU Lockup - switching to software fbcon  Update 2: If I try to install with the regular version of Ubuntu, the screen goes staticy. This also happens if I install with Wubi (distasteful I know) and then try to use Ubuntu.  Update 3: I've tried xforcevesa, nomodeset and xforcevesa nomodeset as my wubi boot options.  Update 4: Thanks to jokerdino, I can now connect to the internet, so internet tools would now be a possibility. Series with denominator that is square of product of consecutive numbers Evaluate $\sum_{k=1}^{\infty} \frac{1}{k^2(k+1)^2}$.   I've tried to use a telescoping approach, but it doesn't seem to work. It doesn't also seem to converge to a "pretty" value as well. $$\displaystyle\int \:\sin^n\left(x\right)\cos^m\left(x\right)\mathrm  dx=\frac{\sin^{n+1}x\cos^{m-1}x}{m+n}+\frac{m-1}{m+n}\int \:\sin^nx\cos^{m-2}x\,\mathrm dx$$  I have been trying to solve for over a week now can someone please help me. Note: This is a canonical post created so that we can close the many similar questions as duplicates as discussed in .   We have created a logo for a client and they love it. We've already started preparing the materials that will feature the new logo.   However, after we had created the logo, we found a logo for another company that looks eerily similar. We did not intentionally create the new logo to look like the one of the other company, in fact we didn't even know it existed beforehand, but it does regardless.  What should we do? Is this plagiarism? Can our client use the logo we have created? Should they? Or do we need to come up with another logo that's more different? Stack Overflow, Stack Exchange, Area 51: What's the difference in a nutshell? Stack Overflow, Stack Exchange, Area 51: What's the difference in a nutshell? I've read the faq's and they all seem to be the same to me - Just Q &amp; A on a variety of topics.  What's the difference in a nutshell? If I have a software engineering question, which site is best for me to use? How to secure recommendations and apply for PhD after having worked outside of academia for two years? In my work, we are comparing predicted rankings versus true rankings for some sets of data. Up until recently, we've been using Kendall-Tau alone. A group working on a similar project suggested we try to use the  instead, and that they preferred it. I was wondering what the differences between the different rank correlation algorithms were.  The best I've found was , which claims Spearman is used in place of usual linear correlations, and that Kendall-Tau is less direct and more closely resembles Goodman-Kruskal Gamma. The data I'm working with doesn't seem to have any obvious linear correlations, and the data is heavily skewed and non-normal.  Also, Spearman generally reports higher correlation than Kendall-Tau for our data, and I was wondering what that says about the data specifically. I'm not a statistician, so some of the papers I'm reading on these things just seem like jargon to me, sorry. EmptyListToVisibilityConverter