Let $$ f(z) = \sum_n z^{n!} $$ I am trying to show that $f(z)$ is analytic on the open unit disk ${|z| &lt;1}$. My thought is to find its radius of convergence $R$, but this series does not have a form of a power series (which I learned how to find $R$ from the various tests) so I am not sure if I am doing it correctly.
I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be greatly appreciated!  (i) $\displaystyle\sum_{n=0}^\infty \bigg(\dfrac{1}{2n}\bigg)z^{3n}$.  (ii) $\displaystyle\sum_{n=0}^\infty(2^n+i)(z-i)^n$.  (iii) Show that the radius of convergence of $\displaystyle\sum_{n=0}^\infty z^{n!}$ is $1$ and that there are infinitely many $z \in \mathbb{C}$ with $|z|=1$ for which the series diverges.
Can the preposition 'in' be omitted here? What is the average number of deaths per episode? While watching Game of Thrones and seeing people being killed left and right, I began to wonder:  Just how many people die per episode (not just major characters but the minor characters as well)?   I'm just wondering about an average, but a list that mentions the number of deaths for each episode would be good too. This is all in iTerm2 on OS X.  I used to have  export TERM='xterm-256color'   in my .bashrc. This meant Vim in tmux did use 256 colors.  And once I added  set -g xterm-keys on   then keyboard shortcuts with modifiers worked fine in Vim. Namely: shift+left/right that I mapped to switch Vim tabs, ctrl+up/down that I mapped to move ("bubble") lines, and shift+left/right which worked out of the box to jump by word in the command-line mode (e.g. when typing something like :e foo bar baz).  However, this setup had the problem that the Vim background color only shows behind text, as mentioned .  So I removed  export TERM='xterm-256color'   from my .bashrc and instead put this in my .tmux.conf:  set -g default-terminal "screen-256color"   That fixed the Vim background color, but broke the keyboard shortcuts - they do unexpected things (move the cursor, delete text) instead.  By using ctrl+v to insert the verbatim output from the key combinations (as described ), I was able to work around it:  map [1;5A &lt;C-Up&gt; map [1;5B &lt;C-Down&gt; map [1;2D &lt;S-Left&gt; map [1;2C &lt;S-Right&gt; cmap [1;2D &lt;S-Left&gt; cmap [1;2C &lt;S-Right&gt;   This makes the shortcuts work, but it doesn't feel like the right solution. Could anyone tell me what's happening here and how to fix it? Breitbart today:   What is the source for the term nothing-burger? C++ namespace and include Why do we need both using namespace and include directives in C++ programs?  For example,   #include &lt;iostream&gt;  using namespace std;  int main() {  cout &lt;&lt; "Hello world"; }   Why is it not enough to just have #include &lt;iostream&gt; or just have using namespace std and get rid of the other?  (I am thinking of an analogy with Java, where import java.net.* will import everything from java.net, you don't need to do anything else.) Applying PCA to data that fail a normality test \mathcal(O) and font size I use the following mystic TeX so as to display the Landau's notations for big and small O.  There is one partial solution to improve the scriptscriptstyle so as to not use a small o : see the secund answer below. The problemen with \smallOBis is the wrong spacing, contrary to \smallO. How can I do this ?  % Sources : %     1) http://forum.mathematex.net/latex-f6/bonnes-commandes-de-base-t12278.html %     2) http://tex.stackexchange.com/questions/30944/mathcalo-and-font-size  \documentclass{article}     \usepackage{amsmath}     \usepackage{amsfonts}     \usepackage{amssymb}      \usepackage{graphicx}      \newcommand{\bigO}[1]{\ensuremath{\mathop{}\mathopen{}\mathcal{O}\mathopen{}\left(#1\right)}}      \newcommand\smallO[1]{         \mathchoice             {% \displaystyle                 \ensuremath{\mathop{}\mathopen{}{\scriptstyle\mathcal{O}}\mathopen{}\left(#1\right)}             }             {% \textstyle                 \ensuremath{\mathop{}\mathopen{}{\scriptstyle\mathcal{O}}\mathopen{}\left(#1\right)}             }             {% \scriptstyle                 \ensuremath{\mathop{}\mathopen{}{\scriptscriptstyle\mathcal{O}}\mathopen{}\left(#1\right)}             }             {% \scriptscriptstyle                   \ensuremath{\mathop{}\mathopen{}{o}\mathopen{}\left(#1\right)}             }     }      \newcommand\smallOBis[1]{         \mathchoice         {% \displaystyle             \scriptstyle\mathcal{O}\left(#1\right)         }         {% \textstyle         \scriptstyle\mathcal{O}\left(#1\right)         }         {% \scriptstyle         \scriptscriptstyle\mathcal{O}\left(#1\right)         }         {% \scriptscriptstyle         \scalebox{0.8}{$\scriptscriptstyle\mathcal{O}$}\left(#1\right)         }     }   \begin{document}  \[     \renewcommand{\arraystretch}{1.2}%     \begin{array}{llll}          \textbf{Style}            &amp; \verb!\bigO!         &amp; \verb!\smallO!                          &amp; \verb!\smallOBis!         \\ \hline         \verb!\displaystyle!      &amp; \displaystyle      \bigO{\frac{1}{n}}         &amp; \displaystyle      1 + \smallO{\frac{1}{n}} - 2 &amp; \displaystyle      1 + \smallOBis{\frac{1}{n}} - 2         \\         \verb!\textstyle!         &amp; \textstyle         \bigO{\frac{1}{n}}         &amp; \textstyle         1 + \smallO{\frac{1}{n}} - 2 &amp; \textstyle         1 + \smallOBis{\frac{1}{n}} - 2         \\         \verb!\scriptstyle!       &amp; \scriptstyle       \bigO{\frac{1}{n}}         &amp; \scriptstyle       1 + \smallO{\frac{1}{n}} - 2 &amp; \scriptstyle       1 + \smallOBis{\frac{1}{n}} - 2         \\         \verb!\scriptscriptstyle! &amp; \scriptscriptstyle \bigO{\frac{1}{n}}         &amp; \scriptscriptstyle 1 + \smallO{\frac{1}{n}} - 2 &amp; \scriptscriptstyle 1 + \smallOBis{\frac{1}{n}} - 2     \end{array} \]  \end{document} Optimizing a screenshot for adding in a document and printing Do you need to disavow on [HTTPS SSL] after performing a disavow on [HTTP] I have a client which disavowed a few thousand dirty links from his site profile back in February 2015. Recently the site was switch from HTTP to HTTPS (SSL enforced).   Google recommends that you add all variations to the Google Console (Webmaster Tools).  The site has a total of 4 variations, eg:    http://www.example.com (Previously Preferred) http://example.com https://www.example.com (Now Preferred) https://example.com   Question:  Since technically this is a new property in Google Webmaster Tools, do you need to submit the same disavow file added previously back in 2015? List creation/manipulation how to sort by the day of the week? I have two files.  file1:  Dave 734.838.9800   Bob 313.123.4567   Carol 248.344.5576   Mary 313.449.1390   Ted 248.496.2204   Alice 616.556.4458      file2:  Bob Tuesday   Carol Monday   Ted Sunday    Alice Wednesday   Dave Thursday     Mary Saturday     I merged the two files.  file3 should look like this:  Name      On-Call     Phone   Carol     MONDAY      248.344.5576   Bob       TUESDAY     313.123.4567   Alice     WEDNESDAY   616.556.4458   Dave      THURSDAY    734.838.9800   Nobody    FRIDAY      634.296.3356   Mary      SATURDAY    313.449.1390   Ted       SUNDAY      248.496.2204     But I cannot get the weekdays to be in order. How do I go about doing that? I am non EU national, living in a Schengen country for past 5 years. I hold Indian passport and EU permanent residence card valid for 5 years.   Can i go to UK for few days to visit friends without a visa?  I am aware that UK is not in Schengen, and Schengen visas and residence permits issued by Schengen country is not valid for United Kingdom? But what about the PR card issued by a Schengen country?  I am also aware that EU citizens carrying EU passport have no restrictions to visit UK. Is it possible to set an SSLProtocol in Apache for a single VirtualHost (poodle)? I'm trying to test a patch for the  that involves disabling SSLv3 on my web server. In order to test this on a non-production environment first, I'm setting the SSLProtocol on a VirtualHost for a different test server. My config looks something like this:  &lt;VirtualHost *:443&gt;     SSLEngine On     SSLProtocol All -SSLv2 -SSLv3     ServerName test.mysite.com     # bunch of other stuff omitted &lt;/VirtualHost&gt;   However even after restarting apache my test site is still claimed to be vulnerable. Is this expected to work? I'm wondering if I have to set it in the global ssl config or whether there's something else subtle that's causing the setting not to take and/or work. PostgreSQL execute a default query when no row are returned I need to calculate the average score of a list of userId.  I have a table user_place_score:  (user_id, place_id, score)   Sometimes I do not have the score (no row for (user_id,place_id) in the table) so I need to fetch a "default" score from the table default_place_score:  (place_id, score)   Is is possible in PostgreSQL to do this in one query? For a given list of users and one place_id, select all their scores in the user_place_score and if no row is found then take the score found if the default_place_score table?  Basically I want to avoid doing this kind of stuff:  for (Integer userId : userIds) {   score = (query to fetch the score for this userId)   if (score == null)     defaultScore = (query to fetch the default score for this PlaceId)    total = total + defaultScore }  //return the average return total/numberOfUser Accourding to ,   We often use "been to" when we refer to completed visits to a place:  Have you ever been to Budapest?  Not: Have you ever gone to Budapest?  If the visit is not complete, we use "gone to".  Joan’s just gone to the shop. She’ll be back in about ten minutes. (Joan has not yet returned from the shop.)  Joan’s just been to the shop. She bought some cakes. Would you like one? (Joan has returned from the shop.)  Now, does that rule apply to the  "Have been on a trip" &amp; "Have gone on a trip"?  So, "I have been on a trip" (I completed the trip &amp; currently I am staying at home)  "I have gone on a trip" (I haven't completed the trip &amp; currently I am on the trip)  And what about "gone crazy" &amp; "been crazy".  Does "He has gone crazy" mean now he is still crazy?  Does "He has been crazy" mean now he is not crazy any more?  Are there any differences between "Have been on a trip" vs "Have gone on a trip" and "Have been crazy" vs "Have gone crazy"?  Note: I am not asking about "been to" or "gone to". I am asking about "been on a trip" &amp; "gone on a trip". Do we apply the same rule of "been to" or "gone to" to "been on a trip" &amp; "gone on a trip"?  Note2:  seems to prove that my thinking is right. Are submodules of finitely generated free modules always free? I don't believe it's true that all submodules of a finitely generated free module are free just based on results from google. Is there a canonical simple example of a finitely generated free module with a submodule that is not free? How to force GPG to use console-mode pinentry to prompt for passwords? How can I disable creepers' block damage on my Minecraft server? Please note that these are NOT acceptable solutions:   Completely disabling monsters (peaceful mode) Completely removing creepers Completely disabling explosions (TNT + Creepers) Disabling creeper explosions   More exactly, I want the creepers to be able to spawn and explode, but without doing their terrain damage (they should still be doing health damage to the players).  I'm fine with the use of mods but I'd prefer a built-in solution if possible. Selecting Unique Elements From a List in C# Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.