$\lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} = L$ implies $\lim_{n\rightarrow\infty}a_n^{1/n}=L$
Convergence of Ratio Test implies Convergence of the Root Test
"The train will leave" vs. "is going to leave" vs. "leaves" vs. "is leaving" I'm porting a module that currently works with Blocks/BlockForms to Paragraphs/NodeForms. I'm inserting an embedded form inside a Paragraph inside a Node Edit page using a hook_field_widget_entity_reference_paragraphs_form_alter. The general structure where the form gets inserted is: $form['field']['widget'][0]['settings'][form here]  After this the form displays as expected on the node edit page. The inserted form goes through a number of #process steps. The element in question is a select form that when the button underneath it is pressed, is supposed to retrieve some data via a custom submit handler and place it on the page with ajax:  The array for this element looks like this:  You can see the custom submit handler that is supposed to fire when 'Select Segment Set' is pressed After debugging, I've found that the entire NodeForm appears to be submitted to the backend, and the only handler that is found is from some other field that uses ajax. In short 'updateSegmentSet' is never discovered as a submit handler. When I debug the original module based on block/blockforms, I do see the correct submit handler being discovered. Now in the latter case, of course a blockForm is submitted to the back end with the correct plugin. Given that this is basically a form within a form within a form, is there something special I have to do to get the custom submit handler to run? I've read a few things online e.g. the 'settings' key and '#tree' = true. EDIT The Ajax property looks like this:       '#ajax' =&gt; [         'callback' =&gt; [$this, 'updateSegmentSetAjax'],         'wrapper' =&gt; $wrapper_id,       ],    public function updateSegmentSetAjax(array &amp;$form, FormStateInterface $form_state) {     $button = $form_state-&gt;getTriggeringElement();     // Go one level up in the form, to the widgets container.     return NestedArray::getValue($form, array_slice($button['#array_parents'], 0, -2));   }  As best I can tell, this is never called either. Are there any rules concerning what I can put in my profile? “All you have to do is read” vs. “All you have to do is to read” I was speaking to an English learner and said, “All you have to do is read a lot.” And they thought that sentence wasn’t grammatically correct because I dropped the word to between is and read.  They thought it should be “All you have to do is to read a lot.” That sounds weird to me.   How do I explain to them the reason you drop the to? Or am I incorrect and it really should have a to? Ok... so this is embarrassing.  I typed in "boobs" on EncyclopediaDramatica's search because, well, it's a really funny page.  But now I've got a problem... EVERY TIME I click a search field in Chrome now, it shows "boobs" as the first suggestion in the drop down!  This is not really socially acceptable as this laptop goes with me everywhere and Chrome is my primary browser.  Is there a way I can isolate and remove just that one entry? Review sections to be always in the same place I'd like to do it in Python. What I'd like to do in this example in C: #include &lt;stdio.h&gt;  int main() {     int i;     for (i=0; i&lt;10; i++) printf(&quot;.&quot;);     return 0; }  Output: ..........  In Python: &gt;&gt;&gt; for i in range(10): print('.') . . . . . . . . . . &gt;&gt;&gt; print('.', '.', '.', '.', '.', '.', '.', '.', '.', '.') . . . . . . . . . .  In Python print will add a \n or space, how can I avoid that? Now, it's just an example, don't tell me I can first build a string then print it. I'd like to know how to &quot;append&quot; strings to stdout. What is a good book to learn number theory? Proofs involving limsup and liminf I've been working with proofs involving $\limsup$ and $\liminf$, and I'm a bit confused regarding their general methodology. More specifically, I'm unsure about whether my approach to the following problem makes sense.  Problem. Let $(s_n)$ and $(t_n$) be sequences and suppose that there exists $N_0$ such that $s_n \leq t_n$ for all $n &gt; N_0$. Show that $\liminf s_n \leq \liminf t_n$ and that $\limsup s_n \leq \limsup t_n$.   The way I approached it was as follows:   Let $N &gt; N_0$. Then $\limsup_{N \rightarrow \infty} \{ s_n : n &gt; N \} \leq t_n$ as $s_n \leq t_n$, and $\limsup s_n $ is the largest possible limit of a subsequence of $s_n$. As $t_n : n &gt; N$ is (by definition) less than $\limsup_{N \rightarrow \infty} \{ t_n : n &gt; N \}$, the proof is complete.  I'm pretty sure this is incorrect, however, and I'm generally unclear about the method behind such a proof. Any help is appreciated! Finding $\int x^xdx$ Featured On Meta section doesn't list MSE posts on meta sites Viewing computer screen through 3D glasses, why the asymmetry? How long do you get to use a UK standard visitor visa from time of issue or is it valid for entry indefinite I am going to apply for a standard visitor visa for my girlfriend to enter UK. If it is issued how soon does it have to be used after issue, ie. validity until time of entry? Zooming to selection in arcpy.mapping and only seeing selected features? I selected data by using select layer by attribute, then I need to only select these selection and zoom in to the selection.  my code is this, it can zoom in base on the scale I set, but I see other adjacent polygon that has other Code_No, why?  Layer = arcpy.mapping.ListLayers(mxd, "lyr", df)        arcpy.SelectLayerByAttribute_management(lyr, "NEW_SELECTION", "Code_NO = '%s'"%mycodeNo)            df.extent = lyrLayer1.getSelectedExtent() df.scale *= 2 $ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$ How can I prove that $\,ax\equiv b\pmod{\! n}\,$ has a unique solution if $\,\gcd(a,n) = 1$?  I suspect that the Bezout identity for $\,\gcd(a,b) = 1$ is the key but I don't see how to proceed. Pgfplots: How to fill area under a curve in a 3d plot similar to \closedcycle in 2d What's the difference between the server version and the desktop version? What's the difference between the server version of Ubuntu and the desktop version? I'm trying to create a pmatrix whose components are rather complicated fractions (Christoffel symbols), and the line spacing is too small making the whole thing a bit cramped and hard to read.  Is there any easy way to increase the spacing (about 1.5 the default would be perfect)? How do I update my TeX distribution? Cartoon with a young unicorn and a demon