I tried to solve but could not develop the left part of the equation    Thanks
a) Give a combinatorial proof that for every $n \geq r \geq 1$ that:  $$\sum_{i = 0}^{n} \binom{i}{r - 1} = \binom{n + 1}{r}$$  And use (a) to concoct a formula for $1^2 + 2^2 + ... + n^2$
$$ \sum_{n=1}^{\infty} \frac{(\sin n+2)^n}{n3^n}$$  Does it converge or diverge?   Can we have a rigorous proof that is not probabilistic?   For reference, this question is supposedly a mix of real analysis and calculus. I'm unable to upgrade Xcode 11GM to 11.1 for some time on my Mid 2014 MacBook Pro running macOS Catalina. I've installed Xcode from Mac AppStore ages ago (IIRC circa 2014), and upgraded step by step since then. Currently, I'm experiencing the following loop:   AppStore reports that I have an update. I open AppStore, open updates, click "Update" on Xcode. Mac Downloads ~2.5GB first. Installs that patch. Continues to get the full package (~7.6GB). It starts to install. After some churning and install, it reports "Cannot download Xcode now, please try again later."   I have more than enough space on the disk, and nothing is modified in Xcode or in the OS (no homebrew or something similar).  Did anyone experience something similar? Are there solutions to this? Is it "legal" to CREATE and DROP #SomeTable more than once? I've always had a LOST.DIR folder on my external SD card, which until now has been empty.  Today my phone complained that my SD card was damaged and wouldn't mount it, so I pulled it out and looked at it on my computer.  There's now a 70237 file inside LOST.DIR which is 128 MB.  It starts with something like "&euml;&lt;MSWIN4.1", seemingly to indicate that it's a dump of a  (which makes no sense).  What did Android dump here and and why?  (I think I may have disconnected my phone from my PC yesterday without ejecting the drives first, so that might be the "why", though it was from Windows XP and not Windows 98.)  Is there a simple way to recover the information in this file? Non abelian Lie group with abelian Lie algebra WOL was working great on Bionic until yesterday's kernel update. How to diagnose/recover it?  Edit: if set to boot from 4.15.0-23 kernel it works. If set to boot from 4.15.0-24 it does not.  Edit2: WoL can be enabled with   sudo ethtool -s [card] wol g   but it isn't persistent. Searching for how to set it every boot.  Edit3: Per a comment by Kai-Heng Feng on this  this was an advertent change! (and therefore not a bug)  This is because the fix of LP: #1752772 has one commit [1] that disables WoL by default. Set WoL explicitly should solve your issue.  [1] https://github.com/torvalds/linux/commit/7edf6d314cd061e1d0a1b7bc0b511d64322c3f72 Misleading chat usercard on SE 2.0 chat Solving $\sin x\cos3x+\cos x\sin3x=\sqrt{3}/2$ in the interval $[0,2\pi]$ Solve the following trig equation: $$\sin(x)\cos(3x)+\cos(x)\sin(3x)=\frac{\sqrt{3}}{2}$$ in the interval $[0,2\pi]$.  Using trig identities, it is now at $$\sin(x+3x)=\sqrt{3}/2 \tag{1}$$ Next, it has become $$\sin(x+3x)=\sin 60^\circ \tag{2}$$ leading to $$\sin(x+3x)-\sin60^\circ=0 \tag{3}$$ From there, it is unclear what to do. Why does a relay have a minimum applicable load? I am trying to find relays for my application and I read  which looks fine but specifies     Minimum applicable load: 10mV 10µA   In my circuit, it is expected that the relay closes but no voltage and current is applied. You can think of it as 2 relays in series where one is open and one is closed, so there is no current. This sounds like something I'd do since I was at school.  Why would a relay require a minimum voltage and current on the load side? Is it allowed to operate that relay under my conditions or not? What could possibly break in a relay if I don't respect this requirement? What does "minimum applicable load" mean? When and how do I need to consider this value? What are the differences between combat damage, direct damage, and loss of life? Is "logicalising" a valid word? Is logicalising a valid word, if the meaning was to make something supposedly 'illogical', logical? ''More than one row returned by a subquery used as an expression'' postgres error There is a badge named Electorate which is going to be awarded on "Voted on 600 questions and 25% or more of total votes are on questions". Can somebody explain this to me?  Voted on 600 Question and 25% are on Question? is this  means   Voted on 600 Posts and 25% are on Question  Can't SO place some easy text or Definitions for average users? I'm quite confused about attacking with two weapons in both hands and wondering what is required to be able to do to (in Pathfinder).  1. Base Attack Bonus Looking at the base attack bonus tables for different classes (e.g. the ) it seems like up to +5 you get only one attack per full action but starting from base-attack-bonus of +6 you get a second attack with +1.  2. Feats There is a feat . The existence of the feat sounds like you could not do two-weapon-fighting without it. However, in the description of "normal" it says:     If you wield a second weapon in your off hand, you can get one extra attack per round with that weapon. When fighting in this way you suffer a –6 penalty with your regular attack or attacks with your primary hand and a –10 penalty to the attack with your off hand. If your off-hand weapon is light, the penalties are reduced by 2 each.   3. Core Rulebook Chapter about combat If I understand the  it says that you need a full-round action do attack with both weapons in two hands.  So how do these things interact? Can I only attack with my off-hand weapon if my base-attack-bonus is high enough that I get second one with +1?  If not, how does the "multiple attacks from high base-attack-bonus" interact with dual-wielding? Assuming I reached a state of base-attack-bonus with +11/+6/+1 and have a full-round action to attack. Does that mean I can either attack with my primary weapon three times or use the primary weapon and off-hand weapon to attack only twice? Why would I then ever choose dual-wielding? Or can I attack three times with both weapons (leading to a total of 6 attacks with heavy malus) Or would I get 3 attacks with the primary weapon + 1 attack with the off-hand weapon? If so, what attack bonus do I have to use for each of the attacks? Random place change in row with restrictions How could I randomly change places of characters in a text file, with some probability (say 10%)?  file:  1  34567 2  12356 3  16987   outcome:  1  35674 2  12653 3  61987 How do I create a script file for terminal commands? Why are objects at rest in motion through spacetime at the speed of light? I read that an object at rest has such a stupendous amount of energy, $E=mc^2$ because it's effectively in motion through space-time at the speed of light and it's traveling through the time dimension of space-time at 1 second per second as time goes forward.  What troubles me here, is the fact that it is traveling through space-time at the speed of light. Why is it at the speed of light? What is the Big-O of a nested loop, where number of iterations in the inner loop is determined by the current iteration of the outer loop? What is the Big-O time complexity of the following nested loops: for (int i = 0; i &lt; N; i++) {     for (int j = i + 1; j &lt; N; j++) {         System.out.println(&quot;i = &quot; + i + &quot; j = &quot; + j);     } }  Would it be O(N^2) still? Quick-and-dirty way to ensure only one instance of a shell script is running at a time What's a quick-and-dirty way to make sure that only one instance of a shell script is running at a given time? I was asked today in class why you divide the sum of square error by $n-1$ instead of with $n$, when calculating the standard deviation.  I said I am not going to answer it in class (since I didn't wanna go into unbiased estimators), but later I wondered - is there an intuitive explanation for this?!