Prove $1! + 2! + . . . + n! < (n + 1)!$ using mathematical induction
Use induction to prove the following: $1! + 2! + .... + n! \le (n + 1)!$
I'd like to colorize a cell based on its contents. 0 = red, 100 = green, and linearly interpolated between.  function LinInt(x){       var ss = SpreadsheetApp.getActiveSheet();   var cell = ss.getActiveRange();   var hue;   hue = (x/100)*120;    var color = HSVtoRGB(hue, 40, 100);   cell.setBackground(color);    return x;     }     function HSVtoRGB(h, s, v) {     var r, g, b, i, f, p, q, t;     if (h &amp;&amp; s === undefined &amp;&amp; v === undefined) {         s = h.s, v = h.v, h = h.h;     }     i = Math.floor(h * 6);     f = h * 6 - i;     p = v * (1 - s);     q = v * (1 - f * s);     t = v * (1 - (1 - f) * s);     switch (i % 6) {         case 0: r = v, g = t, b = p; break;         case 1: r = q, g = v, b = p; break;         case 2: r = p, g = v, b = t; break;         case 3: r = p, g = q, b = v; break;         case 4: r = t, g = p, b = v; break;         case 5: r = v, g = p, b = q; break;     }   return '#'+((b | g &lt;&lt; 8 | r &lt;&lt; 16) / 0x1000000).toString(16).substring(2); }   When I do this I get an error saying that I am not allowed to call setBackground. I assume this is because I'm not allowed to change the color of one cell from another because I'm screwing up getting the current cell that the function is running in.  How do I get the cell that the function is running in so that I can call setBackground() on it?  For example, I'd like to type =LinInt(50) into cell A1 and have A1 be yellow and have the number 50 in it. Then when I type =LinInt(100) into cell A2 it is green and has the number 100 in it. How to correctly add a path to PATH? Pokemon Go login issues after update I downloaded the July 30th update for Pokemon Go and when I open the app I'm prompted to login, But once I login, it redirects me to this page on Cookies::     If I click "Done" in the top left, it prompts me to login again! I've tried reinstalling the game, restarting my phone, etc. I can't much enable Cookies on the inline Safari browser. It doesn't make sense, and cookies are on in Safari (because I don't use Safari and they're enabled by default).  How can I fix this? What is Chromatic Aberration? What’s the correct URL for placing a call on an iPhone? With a macro such as:  \def\naive{na\"{\i}ve}   I find that the space which I would hope to follow it, is absent. So, with  the \naive approach   I get     the naïveapproach   in the output. How do I bring back the space? Getting "experimental" true displacement to work I have a graphic on the clipboard on OS X.  What's the quickest way to get it onto disk as a png or jpg? Is there a way to exclude other particle systems from the effects of a Wind effect? I have a scene with a number of particle systems. I'd like to have a separate wind effect for my main scene and another for an extra: an emitter + wind-effect to blow leaves into my scene. I can't find any way to exclude, so here I am. At the very worst, I can just render the main scene and then render the wind effect and composite it, but I figured I'd check first. Family in GLM - how to choose the right one? When modeling data sampled in the field, I often come across the problem of determining the Family of the dependent variable for GLM (or GLMM). An example: in an ecological study, I have ~ 60 patches.  I have sampled territories of an animal species within these patches as well as a set of explanatory variables.  The number of territories per patch (NTP) is the dependent variable in a GLM.   NTP is integer with values between 0 and 14:      How can I test for the family of NTP?  Any help appreciated as this is an issue many ecologists come across each time they conduct statistical analysis. Why doesn't C# support implied generic types on class constructors? C# doesn't require you to specify a generic type parameter if the compiler can infer it, for instance:  List&lt;int&gt; myInts = new List&lt;int&gt; {0,1,1,     2,3,5,8,13,21,34,55,89,144,233,377,     610,987,1597,2584,4181,6765};  //this statement is clunky List&lt;string&gt; myStrings = myInts.     Select&lt;int,string&gt;( i =&gt; i.ToString() ).     ToList&lt;string&gt;();  //the type is inferred from the lambda expression //the compiler knows that it's taking an int and  //returning a string List&lt;string&gt; myStrings = myInts.     Select( i =&gt; i.ToString() ).     ToList();   This is needed for anonymous types where you don't know what the type parameter would be (in intellisense it shows up as 'a) because it's added by the compiler.  Class-level type parameters don't let you do this:  //sample generic class public class GenericDemo&lt;T&gt;  {     public GenericDemo ( T value )      {         GenericTypedProperty = value;     }      public T GenericTypedProperty {get; set;} }  //why can't I do: int anIntValue = 4181; var item = new GenericDemo( anIntValue ); //type inference fails  //however I can create a wrapper like this: public static GenericDemo&lt;T&gt; Create&lt;T&gt; ( T value ) {     return new GenericDemo&lt;T&gt; ( value ); }  //then this works - type inference on the method compiles var item = Create( anIntValue );   Why doesn't C# support this class level generic type inference? My international conference in Europe this May has been canceled due to the COVID-19. I have a ticket, purchased from TAP air Portugal, from Toronto, Canada to Lisbon. Because it is a non-refundable fare, they don't want to give a refund.  They won't allow me to reschedule later than May 31 of this year. This won't be useful for me.  What options do I have? I'd prefer to get a full refund.  (I am a US citizen if it matters)  Note: They offered to refund just the taxes, which is about 10% of the value. That's not horrible, but I'd rather have the whole thing returned. I'm having trouble understanding Cantor's diagonal argument.  Specifically, I do not understand how it proves that something is "uncountable".  My understanding of the argument is that it takes the following form (modified slightly from the  article, assuming base 2, where the numbers must be from the set $ \lbrace 0,1 \rbrace $):  \begin{align} s_1 &amp;= (\mathbf{0},1,0,\dots)\\ s_2 &amp;= (1,\mathbf{1},0,\dots)\\ s_3 &amp;= (0,0,\mathbf{1},\dots)\\ \vdots &amp;= (s_n \text{ continues}) \end{align}  In this case, the diagonal number is the bold diagonal numbers $(0, 1, 1)$, which when "flipped" is $(1,0,0)$, neither of which is $s_1$, $s_2$, or $s_3$.  My question, or misunderstanding, is: When there exists the possibility that more $s_n$ exist, as is the case in the example above, how does this "prove" anything?  For example:  \begin{align} s_0 &amp;= (1,0,0,\mathbf{0},\dots)\ \ \textrm{ (...the wikipedia flipped diagonal)}\\ s_1 &amp;= (\mathbf{0},1,0,\dots)\\ s_2 &amp;= (1,\mathbf{1},0,\dots)\\ s_3 &amp;= (0,0,\mathbf{1},\dots)\\ s_4 &amp;= (0,1,1,\mathbf{1},\dots)\\ s_4 &amp;= (1,0,0,\mathbf{1},\dots)\ \ \textrm{ (...alternate, flipped } s_4\textrm{)}\\ s_5 &amp;= (1,0,0,0,\dots)\\ s_6 &amp;= (1,0,0,1,\dots)\\ \vdots &amp;= (s_n \text{ continues}) \end{align}  In other words, as long as there is a $\dots \text{ continues}$ at the end, the very next number could be the "impossible diagonal number", with the caveat that it's not strictly identical to the "impossible diagonal number" as the wikipedia article defines it:     For each $m$ and $n$ let $s_{n,m}$ be the $m^{th}$ element of the $n^{th}$ sequence on the list; so for each $n$,      $$s_n = (s_{n,1}, s_{n,2}, s_{n,3}, s_{n,4}, \dots).$$      ...snip...      Otherwise, it would be possible by the above process to construct a sequence $s_0$ which would both be in $T$ (because it is a sequence of 0s and 1s which is by the definition of $T$ in $T$) and at the same time not in $T$ (because we can deliberately construct it not to be in the list). $T$, containing all such sequences, must contain $s_0$, which is just such a sequence. But since $s_0$ does not appear anywhere on the list, $T$ cannot contain $s_0$.      Therefore $T$ cannot be placed in one-to-one correspondence with the natural numbers. In other words, it is uncountable.   I'm not sure this definition is correct, because if we assume that $m = (1, \dots)$, then this definition says that "$s_n$ is equal to itself"&mdadsh;there is no "diagonalization" in this particular description of the argument, nor does it incorporate the "flipping" part of the argument, never mind the fact that we have very clearly constructed just such an impossible $T$ list above.  An attempt to correct the "diagonalization" and "flipping" problem:  $$s_n = (\lnot s_{m,m}, \lnot s_{m,m}, \dots) \quad \text{where $m$ is the element index and} \quad\begin{equation}\lnot s_{m,m} = \begin{cases}0 &amp; \mathrm{if\ } s_{m,m} = 1\\1 &amp; \mathrm{if\ } s_{m,m} = 0\end{cases}\end{equation}$$  This definition doesn't quite work either, as we immediately run in to problems with just $s_1 = (0),$ which is impossible because by definition $s_1$ must be $ = (1)$ if $s_1 = (0)$, which would also be impossible because...   Or more generally, with the revised definition there is a contradiction whenever $n = m$, which would seem to invalidate the revised formulation of the argument / proof.  Nothing about this argument / proof makes any sense to me, nor why it only applies to real numbers and makes them "uncountable". As near as I can tell it would seem to apply equal well to natural numbers, which are "countable".  What am I missing? Regression: Transforming Variables When transforming variables, do you have to use all of the same transformation? For example, can I pick and choose differently transformed variables, as in:   Let, $x_1,x_2,x_3$ be age, length of employment, length of residence, and income.  Y = B1*sqrt(x1) + B2*-1/(x2) + B3*log(x3)   Or, must you be consistent with your transforms and use all of the same? As in:   Y = B1*log(x1) + B2*log(x2) + B3*log(x3)    My understanding is that the goal of transformation is to address the problem of normality. Looking at histograms of each variable we can see that they present very different distributions, which would lead me to believe that the transformations required are different on a variable by variable basis.   ## R Code df &lt;- read.spss(file="http://www.bertelsen.ca/R/logistic-regression.sav",                  use.value.labels=T, to.data.frame=T) hist(df[1:7])      Lastly, how valid is it to transform variables using $\log(x_n + 1)$ where $x_n$ has $0$ values? Does this transform need to be consistent across all variables or is it used adhoc even for those variables which do not include $0$'s?  ## R Code  plot(df[1:7]) How do I access previous promise results in a .then() chain? Leave Convert to Edit button enabled all the time in the moderator flag list In the moderator flags list, there is a &quot;Convert to Edit&quot; button on answers which are posted by the person who posted the question. This button is disabled at times, and I am not sure that the algorithm used is useful. My proposal is that the Convert to Edit button in the flag list should always be enabled, as it seems fairly frequent that it's disabled when it would be useful. Justification When the button is disabled, it is required that we pop open the answer view in the flag list, and use the regular Mod menu and choose convert to edit there. When we do that, however, we experience this issue:  (that should also be fixed, by the way!) Here are two items currently in the queue:    Note that one has the button enabled, while the other does not. The only differences I can see in those two answers is the age and the number of answers. Neither question has an accepted answer, but the one with the button disabled only has that single answer. I've only just now noticed that difference with these two posts, so I'm not 100% sure if that's the determining factor. 1 It is perhaps coincidental, but these buttons being (dis)abled are reverse of the action needing to be taken. The one with the disabled button needed to be converted to an edit; the one with the enabled button was an attempt to answer their own question. If I'm right about the determining factor here (that the answer is the only one), I'm not sure that's useful. A good bit of the time, an asker will post an answer due to people requesting additional information. In such cases, it seems like it will be likely there are no other answers. 1  Note 1: I was looking to create a userscript to enable these buttons, and I found something that I think is related as to the reason. It looks like the ones which are disabled are on answers which have comments on them. I noticed the title attribute of those buttons says, &quot;There are comments on the post. Please review&quot;. That does seem to make some sense in theory. However, it still seems like I am often converting them, anyway. With the above-linked, separate issue regarding how the 'mod' link on the expanded view works, this is still a bit of an impediment to our workflow. It occurred to me to check if the button was reenabled once you expanded the view, but it does not. Maybe that's an alternate solution. My plan for my userscript is to enable the button, but change the text color (and leave the title attribute), to point out that there are comments. If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$ Did not find this from this website...  If $$ \gcd(a,b)=1,$$ then there exists integers $x$ and $y$ such that $$xa+yb=1.$$  Now, the tip is to use particular corollary, that states:  The class $[m]_{n}$ generates $\mathbb{Z}/n\mathbb{Z}\Leftrightarrow \gcd(m,n)=1.$  I am totally lost with the corollary.  Let's assume that $\gcd(m,n)=1$. Then, $[m]_{n}$ generates $\mathbb{Z}/n\mathbb{Z}$.  OK! Then what?  There is also follow up, where I have to prove the converse. I am familiar with Bezout's lemma. How do I send an HTTP request in C# without using System.Net classes? Rendering low and res/high res images Is it possible to render in Blender with reduced low res files and then ask Blender to render the final render with 4k high res files? Low and high res files would be numbered identically but would be located in different folders.  Many thanks Project point onto line in Latitude/Longitude Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D.  Diagram: