How to check whether $$\sum_n \frac{(-1)^n (2n)!x^n}{(n!)^2 4^n}$$ converges or diverges. The ratio test is inconclusive and I don't know what to do with $(2n)!$
Determine when $\sum_1^\infty \frac{(2n)!x^n}{n(n!)^2}$ converges. By ratio test, when $|x|&lt;1/4$, the sum converges, $|x|&gt;1/4$ diverges. But I'm not sure about $|x|=1/4$. By Stirling's approximation $\frac{(2n)!(1/4)^n}{n(n!)^2}\sim\frac{4^n}{n{4^n}\sqrt{ \pi n}}\sim\frac{1}{\sqrt{\pi}n^{3/2}}$, so when $|x|=1/4$, the sum converges. But I haven't learned Stirling's approximation yet, so how to prove that when $|x|=1/4,$ the sum converges, without using it?
Parsing JSON from URL List of generally believed conjectures which cannot all be true There are some conjectures which most leading experts believe in albeit no one can prove it yet. For example: $\mathcal{P} \neq \mathcal{NP}$, the Riemann hypothesis or the Collatz conjecture.  My question is whether there is a set of such conjectures which cannot all be true, eg. is a statement such as "Either $\mathcal{P} = \mathcal{NP}$ or the Riemann hypothesis is wrong" true? E: Malformed entry 54 in list file /etc/apt/sources.list (Component) Can space expand with unlimited speed? just after the Big Bang and before inflation the currently observable universe was the size of a coin. One millionth of a second later the universe was the size of the Solar System, which is an expansion much much faster than speed of light. Can space expand with unlimited speed? href to an image label - how to jump to the image instead of the caption below the image? Write access without read access Is it possible for a user to have a write access to a file and not be able to read it? How is it possible?   I tried the following commands:  debianbox@debian:~/posix/io$ touch filetest debianbox@debian:~/posix/io$ ls -l filetest -rw-r--r-- 1 debianbox debianbox 0 14 oct.  03:10 filetest  debianbox@debian:~/posix/io$ echo "Hello World" &gt; filetest debianbox@debian:~/posix/io$ cat filetest Hello World  debianbox@debian:~/posix/io$ chmod u-r filetest debianbox@debian:~/posix/io$ cat filetest cat: filetest: Permission forbidden  debianbox@debian:~/posix/io$   As you can see here, I have write but not read access on this file. How can this be possible? Is this considered as a bug? If not, in what situation would this be useful? Accessing Email logs using REST or SOQL I recently purged Libreoffice because it was buggy.  I then reinstalled the latest version from apt and now it's missing the symbols above the equation editor.  Here is the current Texmaths equation editor:   There used to be shortcuts for latex including frac{}{} and various set theory operations, as well as the binomial distribution on the top of the window.  What happened to them and how do I get them back? I just figured that they were in some package and installed all packages useing sudo apt install libreoffice*, but this didn't help after I refreshed the program.  EDIT 0: I'm using Kubuntu 19.04  sarah@LesserArk:~$ sudo apt-cache policy libreoffice-texmaths libreoffice-writer texlive libreoffice-texmaths:   Installed: 0.43-2   Candidate: 0.43-2   Version table:  *** 0.43-2 500         500 http://us.archive.ubuntu.com/ubuntu disco/universe amd64 Packages         500 http://us.archive.ubuntu.com/ubuntu disco/universe i386 Packages         100 /var/lib/dpkg/status libreoffice-writer:   Installed: 1:6.2.2-0ubuntu2   Candidate: 1:6.2.2-0ubuntu2   Version table:  *** 1:6.2.2-0ubuntu2 500         500 http://us.archive.ubuntu.com/ubuntu disco/main amd64 Packages         100 /var/lib/dpkg/status texlive:   Installed: 2018.20190227-1   Candidate: 2018.20190227-1   Version table:  *** 2018.20190227-1 500         500 http://us.archive.ubuntu.com/ubuntu disco/universe amd64 Packages         500 http://us.archive.ubuntu.com/ubuntu disco/universe i386 Packages         100 /var/lib/dpkg/status Does adding two linear equations will result in a line which will pass through an intersection of the linear equations? I was wondering why it is almost impossible to find a geometrical explanation of why adding two linear equations helps us to find a solution of a system of linear equations? Am I right that adding two linear equations will result in an equation of a line which will pass through a point where two linear equations intersect?  If it is right then I completely don't understand why such a crucial point in understanding how to solve systems of equations is never properly taught even at a university level. Some of the Puella Magi seem to have magical powers related to the wishes that they made. Sayaka has regeneration abilities, and she wished to heal someone. Homura has time manipulation abilities, and she wished to go back in time.  But other magical girls, like Mami and Kyouko, don't appear to have powers related to their respective wishes.  So, is there a connection, or is it only a coincidence for Sayaka and Homura? If it's the former, how do Mami and Kouko's wishes relate to their powers? Can the determinant of a matrix can be made $0$? An entry of an $n \times n$ matrix with nonzero determinant is defined as interesting if by changing this entry (and only this entry) the determinant of the matrix can be made $0$.   Is it true that each entry of every matrix with nonzero determinant is interesting? Is it true that there is an interesting entry in each row of a matrix with nonzero determinant? Proof that in a field $x=0$ is equivalent to $x=-x$ Why is sudoers NOPASSWD option not working? Loading custom configuration files I know I can open config files that are related to an assembly with the static ConfigurationManager.OpenExe(exePath) method but I just want to open a config that is not related to an assembly. Just a standard .NET config file. Why do backslashes appear twice? Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal? What are the strategies for getting feedback on articles? As an academic, one has to publish. Often, an article is drafted, and you need feedback of a colleague.  I see that quite often some perspective articles are often written by long-term collaborating pairs of authors.  What are some strategies people use to get feedback on their articles  (outside of immediate boss; and in the case that none of the friends in the field of work in the specific domain of the article, and one needs specific (not general) feedback)?   How do you approach a colleague to simply read your article. Or do you just mention it at a conference to the most suitable colleague - would you like to read an article and give me feedback? How do handle the co-authorship or acknowledgement? Do you establish the limits at the "approach time"?  What are some strategies to establish a "publishing" duos (buddies). Openlayers v4.0.1 support Google Maps Javascript API? I want to know if the latest version of Openlayers (v4.0.1) supports Google Maps as a Tile Layer. I can't find any document ion about this. If Openlayers doesn't support Google Maps, could someone  tell me if there is any way to do this? Why are 15 reputation points required to upvote? I think more reputation points should be required to add a new answer than to upvote an existing answer.  I don't continually watch for new questions, so usually when I find a question it has already been answered.  It seems bit inappropriate to answer an already-answered question in order to contribute to the conversation, but until you earn reputation, you can't upvote or comment; you can only add new answers, which prevents me from contributing at the moment as much as I would like to.  This seems backwards to me. If someone incorrectly upvotes a bad answer, the damage is minimal, as other users can down vote it and upvote the correct answers. On the other hand, an incorrect answer wastes the time of everyone who reads it.  So, IMHO, it should require more reputation points to add an answer than to upvote or comment on an existing answer.  I wouldn't say say that users should be prevented from answering unanswered questions until they have x reputation points, as that might prevent users who have answers from providing them, but perhaps it should be required if the question is already answered, to discourage users from reiterating previously given answers.  What do you think? Python "extend" for a dictionary