To show that $$\sum\limits_{n=2}^{\infty}\frac{(n^3+1)^\frac13-n}{\log n}$$ is convergent My attempt: Expanding the first term using binomial series and simplifying, I tried comparison test with $b_{n}=\frac{1}{n^2}$  $$\lim_{n \to \infty}\left(\frac{(n^3+1)^\frac13-n}{\log n}\right)n^2 $$ $$=\lim_{n \to \infty}\frac{n\left(\frac13\frac{1 }{n^{3}}-\frac19\frac{1 }{n^{6}}+\dots\right)n^2}{\log n}$$ I could simplify this using binomial theorem but I don't know how to deal with the $\log$ term in the denominator
Show that the series $$\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$$ converges.   I showed it using Abel's theorem and limit comparison test. Any other simpler method?
UISearchDisplayController - how to preload searchResultTableView $\mathbb Z[1/2]$ is not finitely generated ?   Maybe I misunderstood, what finitely generated means.  it says, we need finitely many elements and I think $1$ and $1/2$ suffices as generators.  But , at the end of Proposition $5.1.4$, from the fact that $\mathbb Z[1/2]$ is not finitely generated it is concluded that $1/2$ is not integral. Can I recover a file from /dev/null? How can I resize a partition with Disk Utility? I have 2 partitions, I reduced the size of one and want to expand the other to fill the now available space.   How do I expand the Lion Partition into the empty grey space beneath Macintosh HD partition? How do electric currents make light? In an incandescent light bulb, electrons flow through wires to the bulb, where is passes through a filament, and the filament lights up.  Does this work because of several high energy electrons losing some of their energy to disturb the electromagnetic field, creating photons? The electrons just leave the filament afterward and are replaced by more high-energy electrons from the negative terminal. Or does it work differently?  As a secondary question, is there a type of light bulb that directly converts electron flow to photons? Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$ Number of permutations of $n$ elements where no number $i$ is in position $i$ How to put a brace on the right, not left, to group cases? How can I get a brace on the right, not the left, to group cases, using cases? (I know actual cases look better with the brace on the left, but the items I need to group are not literally cases. They are a set of steps in an algorithm which I want to consider as an item.)  The following puts the brace on the left. All I need to do is to flip it to the right.  \begin{align} \begin{cases} a&amp;=bbb\\ c&amp;=ddddddd\\ e&amp;=ffffffffff \end{cases} \end{align}   Thanks for any help with this! How can I add "page # of ##" on my document? I want to add a footer to my document displaying the current number out of total number of pages. How do I do that? I am having trouble adding an image picker to a block form defined in a custom module.  If I create a custom block type I can define an image field as entity_reference and in the block's form view tell it to use entity_browser and media_entity_browser.  This provides me with an picker with which I can select my image:    I am trying to recreate this in a block provided by my custom module, without much success  // /src/Plugin/Block/MyCustomBlock.php  function blockForm($form, FormStateInterface $form_state)  {     ...      $form['image'] = [       '#type' =&gt; 'entity_browser',       '#entity_browser' =&gt; 'media_entity_browser',       '#title' =&gt; $this-&gt;t('Image'),       '#default_value' =&gt; $this-&gt;configuration['image'] ?? null,       '#weight' =&gt; '0',     ];      ... }   This gives me the following.  Note the lack of the bounding box, title and option to edit remove items:     Exporting the configuration files for the working form view gets me this:  // core.entity_form_display.block_content.lead_block.default.yml      ...  field_image:     weight: 1     settings:       entity_browser: media_entity_browser       field_widget_display: rendered_entity       field_widget_edit: true       field_widget_remove: true       selection_mode: selection_append       field_widget_display_settings:         view_mode: default       open: false     third_party_settings: {  }     type: entity_browser_entity_reference     region: content   ...   But if I try and use 'type' =&gt; 'entity_browser_entity_reference' in my blockForm function the field disappears altogether.  Looking into various core files I can see that entity_browser is a @FormElement and entity_browser_entity_reference is a @FieldWidget but I don't really understand the difference.  As you might guess I am fairly new to Drupal and I feel like I am missing something fundamental.  I (think I) understand that custom block types are actually nodes internally and that they are subject to different rules to blocks which I define in my custom modules, but I don't fully grasp what the differences here are.  Is is it possible to use the entity browser in my custom module block form in the same way as it is used in the form which I build through the admin's custom block type builder?  Thank you. Currently, for all my commands that require root access, I have to type them out on a command line.  I would like for the GUI to prompt me to key in my password, when sudo is required, instead of having to type out the commands from command line.  Is there any way to enable this? Previously one version of Ubuntu desktop had this feature out of the box.  This version of Ubuntu (that I am now running) I upgraded from a server installation.  I will add, my user account is already added to sudo group. Thus I can run commands such as sudo gedit or sudo apt-get but I'm not able to install software in the Ubuntu Software Center. I have got this problem on limits  $$ \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$$ What I am doing here is that taking log and then applying L-Hospital's Rule $$y= \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2} $$ then $$\ln(y)=\lim_{x \rightarrow 0} \frac{1}{x^2}\ln\left(\frac{\sin(x)}{x}\right)$$ now it becomes $\frac{0}{0}$ type , but problem is how to take derivative here? when differentiating $\ln\left(\frac{\sin(x)}{x}\right)$ it will become $\frac{x}{\sin(x)}$ then we'll have to differentiate one more time like we do in case of $f(f(x))$ ? $\sin(x)$ and $x$ will be separately differentiated or together ? How do I create a deb package for a single python script? I have a single python script that I want to distribute as a deb package. It is a  which shows local date in Unity panel. I did follow  but I could not create a deb package because it fails.   Can someone give me a step-by-step instruction on what I should do? As far as I know this script depends on python-appindicator.   NOTE: I do not want any links to Debian/Ubuntu packaging instructions. I have seen most of them. I don't find them beginner friendly. I am new to Linux and Ubuntu. When I don't know how to do something with this OS I search on the web and Ask Ubuntu and always find the answer.  But sometimes it's not a great answer nor complete tutorial. I follow the instructions anyway, but I don't know what all the commands mean.  So my question is:   When I follow a tutorial, am I 100% safe? If not, then how can I tell I am in danger? I upgraded from 17.04 to 17.10 on a Lenovo T540p which has broken two finger touchpad scrolling and right clicking Trackpad two finger scrolling and right-click issue with Ubuntu 17.10 (i386, 64bit, GNOME 3.26.1) on a Lenovo T540p (the model with a Intel Core i5-4210M Haswell CPU etc) Renaming attributes/fields in shapefile attribute table using QGIS I would like to rename some fields in my attribute table:  Like rename &quot;XRF_N3&quot;in something else. It seems quite a easy issue but I don't find any information about how can I do that. $\lim_{x \rightarrow \infty} \lim_{n \rightarrow \infty} f_{n}(x) = \lim_{n \rightarrow \infty}\lim_{x \rightarrow \infty}f_{n}(x) $ in case of uniform convergence of $f_{n}(x)$ to $f(x)$  I used this while I was given that $f_{n}(x)$ converges to $f(x)$ uniformly. Is this correct and I was also thinking to prove this! I recently installed Ubuntu 12.10 and it requires a passphrase to boot up (I installed it with an encrypted file system).  Do I have to reinstall to change to a standard unencrypted file system? Keep the same field headers after joining a table to a shapefile in QGIS I noticed that after joining a table to a shapefile the headers of each fields in the table used for the join are changed by QGIS and they are attributed to a different name.  For instance, I had a shapefile with vegetation_codes and a table with vegetation_codes and veg_definition (these were the headers). After joining the table to the shapefile and also saving the shapefile and creating a new one, the vegetation_classes header was changed into vegetati_1.  Is there a way to keep the original headers?  I am using QGIS Dufour on Windows 7 and the table that I join to the shapefile is CSV (tried with dbf as well and the problem is the same). Matlab arrays operation