I see both type of coding style in use: \newcommand{\hello}{Hello, World}  \newcommand\hello{Hello, World}  My question: Is there any situation where removing the braces around the command name being defined with \newcommand or \renewcommand would provide a result or behavior that is different from what it would if we do not remove the braces around the command?
I've seen in every introductory material on LaTeX and in a lot of code of other people (event experts) that the \newcommand command is used by wrapping the first argument in braces, even if it seems unnecessary to me.   The braces are consistent with the usual way arguments are passed to LaTeX commands (since usually the braces are used even if the argument is made of a single token, for consistency and clarity), but in this single case they appear very ugly to me and in my opinion they lower readability.  But these are opinions, while the question is technical: is there any technical reason why one should write this:  \newcommand{\mycmd}[1]{Hello #1}   instead of this:  \newcommand\mycmd[1]{Hello #1}
Use induction to prove the following: $1! + 2! + .... + n! &lt; (n + 1)!$  Base case:  $n = 1$  $1! &lt; 2!$ true   Inductive step: Assume that $1! + 2! + .... + k! \le  (k + 1)!$ is true   let $n = k + 1$  $1! + 2! + .... + (k + 1)! &lt; (k + 2)!$  $1! + 2! + .... + k! + (k + 1)! &lt; (k + 2)!$  $1! + 2! + .... + k! + (k + 1)! &lt; (k + 2)!$  $(k + 1)! + (k + 1)! &lt; (k + 2)!$  $2(k + 1)! &lt; (k + 2)!$  $2(k + 1)(k)(k - 1).. &lt; (k + 2)(k + 1)(k)(k - 1)....$  $2 &lt; (k + 2)$   the right is always bigger for k are non negative integer. Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$ Today my lecturer put up on the board that:  If $\lim\limits_{n\to\infty}\frac{a_{n+1}}{a_n}$ exists and $a_n&gt;0$ then   $\displaystyle \limsup\limits_{n\to\infty}\left(a_n^{\frac{1}{n}}\right)=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$  however I am not sure why this is true, can somebody give me a hint or something as to how to go about proving this.  thanks for any help How do I make my own horse follow me? I've found a horse in Skryim and I ride it a bit to explore the wild but when I dismount, it wanders away and doesn't stay where I leave it. How do I stop it from doing this? Is there a special thing I forgot about horses? I didn't steal this horse - it was from an Imperial Fort that I've cleared. Web-mapping with editing capabilities? How to remove the left part of a string? I have some simple python code that searches files for a string e.g. path=c:\path, where the c:\path part may vary. The current code is:  def find_path(i_file):     lines = open(i_file).readlines()     for line in lines:         if line.startswith("Path="):             return # what to do here in order to get line content after "Path=" ?   What is a simple way to get the text after Path=? How do I disable "Workspace dock" (a bottom panel with the list of open windows and a workspace indicator-cum-switcher), the one that is over "Plank" in the screenshot below?    I'm using Ubuntu 17.10. Your UK visa application has been issued.      PLEASE DO NOT GO TO THE VISA APPLICATION CENTRE BEFORE THEY CONTACT YOU.      You will be contacted by the Visa Application Centre when your documents are ready for collection, or have been dispatched via courier where this is relevant.       You are reminded that you should avoid making travel plans until you have collected your passport, as the outcome of your visa application is not final until you receive your passport.      Please ensure you check the details on your visa when collecting your passport. If you need to contact us you can do so via our International Enquiry Service at www.gov.uk/contact-ukvi-outside-uk      Kind regards,      UK Visas and Immigration   British Embassy Abu Dhabi     Your UK visa application has been issued.   I know words like processed, forwarded, received, refused but never heard anything like that, so I am confused. Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46. Attempted proof based on  and .  Given: $a,b$ elements of $G$, and $ab$ has finite order $n$.   Hence $\color{magenta}{|ab| = n} \iff (ab)^n = e$.   I must prove that $n$ is the smallest positive integer        such that $(ba)^n = e.$ Notice:  $\begin{align}(\color{darkorange}{a}\color{darkcyan}{b})^n &amp;= e \implies \color{darkorange}{a}\color{darkcyan}{b}(ab)^{n-1} = e \implies  \color{darkcyan}{b}(ab)^{n-1}\color{darkorange}{a} = e \implies (ba)^n = e \end{align} $ $\implies |ba| \le n$.  Now I must prove that there's no positive integer $m$ such that $m&lt;n$ and $(ba)^m = e.$ For a proof by contradiction, suppose $|ba|&lt;\color{magenta}{n}$. Then we could apply the same reasoning to find that $ |ab| ≤ |ba| &lt; \color{magenta}{|ab|}$,  which is absurd. So $|ba| = |ab| = n. \quad \blacksquare$  Questions   Whence does $ |ab| ≤ |ba| $ arise, in the second last line?   What's the intuition of the claim proved?    How do you prognosticate that $|ba|&lt;\color{magenta}{n}$ is impossible, and that you should prove it by contradiction? Suppose we have a sample of size n= 9, where the sum equals 27, and the sum of the squaredvalues equals 113. Sample mean is 27/9=3 and mean of squared values is 113/9 = 12.56 Can we then say that the sample variance $S^2 = E(\overline X^2) - E(\overline X)^2 = 12.55 - 9 = 3.56$ ? What use are Steam Trading Cards? Is it purely a 'gotta-have-them-all' mindset, or do they actually serve some purpose?  I notice that you are able to sell them on a community market. For a test I listed one, and someone bought it for 50p. Not exactly a huge amount, but I don't understand why someone would spend real money on them.  Am I missing something?   Are they redeemable for games or money off vouchers or something 'tangible' that I'm missing? Let$\ p_n$ be the$\ n$-th prime. Can you give me a proof for$\ \prod_{i=1}^\infty \frac{p_i-1}{p_i}=P\approx \frac{1}{11.0453}$? I found$\ \prod_{i=1}^\infty \frac{p_i}{p_i-1}\approx 11.0453$ on . Moreover, writing a paper, should one provide a proof or it is trivial? Thanks in advance. The passive with "let" Does a passive Perception (Wisdom) check add WIS mod + Perception skill? I'm new to D&amp;D. There seems to be a discrepancy between the 5e Starter Set Rulebook and the associated character sheets. The Rulebook says that a passive wisdom (percep) score is "10 + the creature's wisdom modifier, as well as any bonuses."  In the example they have a 1st-level character (with a proficiency bonus of +2) Wisdom of 15 (+2) and a  proficiency in Perception, he or she has a passive Wisdom (Perception) of 14 (10+2+2).  But the Human Fighter Folk hero character has a WIS mod of +1, and a +3 written beside perception, and a passive Wis percep of 13 (not 14?).  The Dwarven Cleric has Wis mod of +3 and a +3 written beside perception, and only a 13 passive wisdom.  I guess this has to do with proficiency in the skill, but I'm confused. Can someone explain these instances and maybe also when proficiency is used, vs when just the modifier or just the skill without proficiency? What determines the direction of precession of a gyroscope? I understand how torque mathematically causes a change to the direction of angular momentum, thus precessing the .  However, the direction, either clockwise or counterclockwise, of this precession seems to me a bit arbitrary beyond mathematical definitions. (What if torque were defined by a "left hand rule", for example?).  What's the fundamental physical reason that the precession of a gyroscope would be directed in a certain direction? I have been working with a string[] array in C# that gets returned from a function call. I could possibly cast to a Generic collection, but I was wondering if there was a better way to do it, possibly by using a temp array.  What is the best way to remove duplicates from a C# array? LWJGL—How to Animate and Render Characters Using a Sprite Sheet I'm making an RPG similar to Super Nintendo RPGs in the early '90s. I'm handling movement much like Squaresoft's Final Fantasy IV.   Notice that the player never actually changes his x and y positions on screen. What I'm asking, to be precise, is how could I, in LWJGL through user input (say, the arrow keys), take the sprite sheet I have of my character's movement animations and render them in a [partially] fluid animation like above?  I've been perusing the internet for hours, and I've not found a surefire way to do this. I don't have any code as of yet; I'm more or less looking for a guideline or some pointers in the right direction concerning how I might tackle this.  And, to note, this'll be a rather large game, so I want to focus partially on speed and efficiency—that meaning that I don't want to dive into Slick2D and libGDX unless absolutely necessary. Can steam create an overview of all my achievements? I play several games on Steam and I love the achievement system. It would be great to get an overview of all the achievements ever gained in Steam games. Does Steam support this? I haven't found such an overview.  If such a feature does not exist, I would already be happy with an easy way to calculate the sum of all my achievements automatically.  Edit: I stumbled upon an interesting website called  today. Unfortunately it only works for XBox-Achievements. Does anyone know of a similar service for Steam? This would solve my question right away. Implementing my own solution based on the links posted in the answers didn't work because the APIs are not sufficient for my needs. Difference of the \dots* What is CUPS server and how to share a printer (Locally or over a network) The title tells you all, except that it's about sharing printers.   How do I connect to a shared printer on the network through CUPS or terminal? Why zero correlation does not necessarily imply independence