I am given the sequence $(x_n)_{n \ge 2}$ such that:  $$x_n = \sqrt[n]{1 + \sum\limits_{k=2}^n(k-1)(k-1)!}$$  And I am asked to find $\lim\limits_{n \to \infty} \dfrac{x_n}{n}$ . While trying to solve this, I first made the following observation:  $$\begin{align}\sum\limits_{k = 2}^n (k-1)(k-1)! &amp;= \\ &amp;=\sum\limits_{k = 2}^n(k-1+1)(k-1)! - (k-1)! \\ &amp;=\sum\limits_{k=2}^nk(k-1)! - (k-1)! \\  &amp;=\sum\limits_{k=2}^nk!-(k-1)! \\  &amp;=2!-1!+3!-2!+4!-3!+5!-4!+...n!-(n-1)! \\ &amp;= \boxed{\color{#D04}{-1 + n!}} \end{align}$$  So, $x_n$ becomes:  $$ \begin{align}x_n &amp;= \sqrt[n]{1-1+n!} \\  x_n &amp;= \color{#a90}{\sqrt[n]{n!}} \end{align}$$  So the limit I have to find is:  $$\lim\limits_{n \to \infty} \dfrac{x_n}{n} = \lim\limits_{n \to \infty} \dfrac{\sqrt[n]{n!}}{n} = \lim\limits_{n \to \infty} \sqrt[n]{\dfrac{n!}{n^n}}$$  And this is where I got stuck, kinda. I could find the limit by using the Stirling Approximation:  $$n! \approx \sqrt{2 \pi n} \bigg (\dfrac{n}{e} \bigg )^n$$  By replacing the $n!$ term in my limit with this approximation, I eventually found that  $$\lim\limits_{n \to \infty} \dfrac{x_n}{n} = \dfrac{1}{e}$$  Which seems to be the right result, according to my textbook.  What I don't like about this is that use of the Stirling Approximation. It always feels a bit off using an approximation to get an exact result. My question is this: is there any other way of arriving at this answer without using the Stirling Approximation? In other words, can I find the following limit:  $$\lim\limits_{n \to \infty} \sqrt[n]{\dfrac{n!}{n^n}}$$  Without using an approximation?
how I can continue this limit resolution?  The limit is:   $$ \lim_{n \to \infty}{\sqrt[n]{\frac{n!}{n^n}}} $$  This is that I have done:  I apply this test: $ \lim_{n \to \infty}{\sqrt[n]{a_n}} = \frac{a_{n+1}}{a_n} $  Operating and simplifying I arrive to this point: $$ \lim_{n \to \infty}{\frac{n^n}{(n+1)^n}} $$  I've done something wrong? Thanks!
In Drupal 8 you can load an entity with:  $node = \Drupal::entityManager()-&gt;getStorage('node')-&gt;load(123);   This looks for the entity definitions and finds that node is defined by Drupal\node\Entity\Node - so (I guess) Drupal\node\NodeStorage will instantiate a new Drupal\node\Entity\Node instance.  What I'd like to achieve is subclassing Drupal\node\Entity\Node and being able to instantiate this subclass when it's appropriate. For example if I have a node bundle article, there would be a class:  namespace Drupal\my_module\Entity\Article; class Article extends Drupal\node\Entity\Node { }   And I'd call:  $node = \Drupal::entityManager()-&gt;getStorage('node_article')-&gt;load(123);   And the return would be my Article subclass.  I can achieve this by creating a new entity type and wiring it back to another existing entity definitions, for example the node-article example would be this class:  namespace Drupal\my_module\Entity; use Drupal\node\Entity\Node; /**  * @ContentEntityType(  *   id = "node_article",  *   label = @Translation("Content"),  *   bundle_label = @Translation("Content type"),  *   handlers = {  *     "storage" = "Drupal\node\NodeStorage",  *     "storage_schema" = "Drupal\node\NodeStorageSchema",  *     "view_builder" = "Drupal\node\NodeViewBuilder",  *     "access" = "Drupal\node\NodeAccessControlHandler",  *     "views_data" = "Drupal\node\NodeViewsData",  *     "form" = {  *       "default" = "Drupal\node\NodeForm",  *       "delete" = "Drupal\node\Form\NodeDeleteForm",  *       "edit" = "Drupal\node\NodeForm"  *     },  *     "route_provider" = {  *       "html" = "Drupal\node\Entity\NodeRouteProvider",  *     },  *     "list_builder" = "Drupal\node\NodeListBuilder",  *     "translation" = "Drupal\node\NodeTranslationHandler"  *   },  *   base_table = "node",  *   data_table = "node_field_data",  *   revision_table = "node_revision",  *   revision_data_table = "node_field_revision",  *   translatable = TRUE,  *   list_cache_contexts = { "user.node_grants:view" },  *   entity_keys = {  *     "id" = "nid",  *     "revision" = "vid",  *     "bundle" = "type",  *     "label" = "title",  *     "langcode" = "langcode",  *     "uuid" = "uuid",  *     "status" = "status",  *     "uid" = "uid",  *   },  *   bundle_entity_type = "node_type",  *   field_ui_base_route = "entity.node_type.edit_form",  *   common_reference_target = TRUE,  *   permission_granularity = "bundle",  *   links = {  *     "canonical" = "/node/{node}",  *     "delete-form" = "/node/{node}/delete",  *     "edit-form" = "/node/{node}/edit",  *     "version-history" = "/node/{node}/revisions",  *     "revision" = "/node/{node}/revisions/{node_revision}/view",  *   }  * )  */ class Article extends Node { }  // Results my Article sub type. $node = \Drupal::entityManager()-&gt;getStorage('node_article')-&gt;load(123);   This works fine (as much as I can see); however, it smells. 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