I know if $\beta = 0$ it converges $\iff$ $\alpha &gt; 1$. Also, it doesn't converge if $\alpha = 0$. I don't know what test to apply for the rest of cases. Any hints?
Study the convergence of the following series: $$\sum_{n=2}^\infty \frac{1}{n^\alpha \cdot\ln^\beta(n)} \text{ where }\alpha,\beta \geq 0 $$   Applying d'Alembert criterion I have that $$ \lim_{n\to\infty} \frac{n^\alpha  \ln^\beta(n)}{(n+1)^\alpha  \ln^\beta (n+1)} = \lim_{n\to\infty} \left(\frac{n}{n+1}\right)^\alpha\left(\frac{\ln(n)}{\ln (n+1)}\right)^\beta = 1$$ so the nature of the series is inconclusive.  If $\alpha = \beta = 0$, then the series diverges, since $\sum_{n=2}^\infty 1 = \infty$.  Should I study the rest of the cases (i.e. if $\alpha = 0, \beta &gt; 0$ the root test and the ratio test are also inconclusive). What is the best form to study the series?.  Thanks in advance
Do the Ents go to the Undying Lands? Badge Name is Blue? Now some random badges on the Profile page for the mobile web is now blue! This is seemingly weird as the default and now permanent color for Badge name text is white. Here's the problem here:     Ahhhh! Now it has invaded the desktop/computer version too! Now seemingly, every version has badges with blue text. Android app has the same issues too! Can the badge color be fixed from blue to white on the mobile version please? Same mesh with different colors Let $R$ be a ring (unital, not necessarily commutative), $M$ a finitely presented left $R$ module. Suppose $m_1,\ldots,m_n\in M$ generate $M$. This determines a surjection $f:R^n\to M$.     Must the kernel of $f$ be finitely generated?   This is true by definition if the generating set $m_1,\ldots,m_n$ is the one coming from the finite presentation of $M$. How to place a float at the top of a floats-only page? How Are Galaxies Receding Faster Than Light Visible To Observers? Proving a Composition of Functions is Bijective If I have $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f$ is onto and $f\circ f\circ f = f$, how can I prove that $f$ is bijective? I know that I only have to prove that it is 1-to-1 because I'm given the fact that it's onto, but how can I use the fact that $f\circ f\circ f=f$ to prove that it is 1-to-1?  Thanks! Why can I not access a file named `-` when quoting? I have a file called -.  I want to display its contents.  One way is to do  cat ./-   since cat - reads from standard input. However, why are cat "-" and cat '-' also interpreted by the shell as cat -? Isomorphism between $\Bbb R$ and $\Bbb R(X)$? My questions are:     $1.$ Is there a field morphism $\Bbb R(X) \hookrightarrow \Bbb R$ ?      $2.$ If the answer to $1.$ is "yes", are $\Bbb R$ and $\Bbb R(X)$ isomorphic as fields?   $ $    $ $  For $1.$, here are some comments:   This is true for $\Bbb C$, see . The key point here is the fact that $\Bbb C$ is algebraically closed, of cardinality $2^{\aleph_0}$, and of characteristic $0$. The same argument doesn't work for $\Bbb R$. We can actually find a subfield $F$ of $\Bbb R$ such that $F \cong F(X) \cong F(X,Y) \cong \cdots$, see .  that $\Bbb Q(X)$ does not embed in $\Bbb Q$ (as a field).  This is equivalent to: Is there an injective ring morphism $\Bbb R[X] \hookrightarrow \Bbb R$ ? (By taking the fraction fields).   As for 2. :   They are isomorphic as abelian groups, and actually as $\Bbb Q$-vector spaces. They are not isomorphic as $\Bbb R$-algebras, because they haven't the same transcendence degree over $\Bbb R$. Let's try to see how to construct such a field isomorphism. We could try$^{[1]}$ to find a subset $B$ of $\Bbb R$ such that $\Bbb R = \Bbb Q(B)$ and $C \subsetneq B \implies \Bbb R \neq \Bbb Q(C)$.  The set $B$ must contain transcendental elements, pick such an $x_0 \in B$.  Then $\Bbb R = \Bbb Q(B) = \Bbb Q(B \setminus \{x_0\})(x_0)$ and we could try$^{[2]}$ to find a field isomorphism $\Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$, so that $\Bbb Q(B \setminus \{x_0\})(x_0) \cong \Bbb Q(B)(x_0) \cong \Bbb R(X)$.  For ${[1]}$, I tried to use Zorn's lemma on $\mathscr B = \{B \subset \Bbb R \mid \Bbb R = \Bbb Q(B)\}$ and the partial order $B_1 ≤ B_2 \iff B_1 \supset B_2$. But this is not clear to me why this should be an inductive set.  For ${[2]}$, my idea was to define $f : \Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$ by choosing a bijection $b : B \setminus \{x_0\} \to B$ (these sets should be uncountable), and then trying to extend $f\vert_{B \setminus \{x_0\}} = b$ to a field morphism.     Edit for $^{[1]}$: in  article (MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS, by LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RUZICKA) the theorem 2.4. proves that $\Bbb R$ has no minimal generating set as field (or $\Bbb Q$-algebra, i.e. as ring). The argument uses Artin-Schreier theorem. Minimal generating sets of modules are   .  I think that $B$ is a minimal generating set of a subfield $K \subset \Bbb R$ over $\Bbb Q$ if and only if any element of $B$ cannot be written as a rational fraction of other elements of $B$. Under these conditions and using (transfinite) induction, we can probably prove (and this was my first idea) that any function $f: B \to \Bbb C$ such that $b$ and $f(b)$ are conjugate over $\Bbb Q$ (i.e. have the same minimal polynomial) extends to a (unique) field morphism $\sigma : K = \Bbb Q(B) \to \Bbb C$.  Actually, the collection $\mathscr B$ defined above doesn't have to be an inductive set, at least if we replace $\Bbb R$ by $K =\Bbb Q(\sqrt 2, \sqrt[4]{2},\sqrt[8]{2},\dots)$, because the descending chain $\left(E_n = \{\sqrt[2^m]{2} \mid m \geq n\}\right)_{n \geq 1}$ satisfy $K = \Bbb Q(E_n)$, but $K \neq \Bbb Q\left(\bigcap_{n≥1} E_n\right)=\Bbb Q$, so the chain has no upper bound w.r.t. $≤$ in $\mathscr B$. Does the harmonic series converge if you throw out the terms containing a $9$? I found this very amusing comic on the internet the other day:   The last frame seems to claim that the harmonic series converges if you throw out all the terms with a $9$ in the denominator. Is this true? Where could I find a proof? Is it only true for terms with a $9$ or is it true for any number? If it's only true for $9$, why is that? Good reasons to prohibit inheritance in Java? What are good reasons to prohibit inheritance in Java, for example by using final classes or classes using a single, private parameterless constructor? What are good reasons of making a method final? Bash shell script output alignment My script:  date echo -e "${YELLOW}Network check${NC}\n\n"  while read hostname do  ping -c 1 "$hostname" &gt; /dev/null 2&gt;&amp;1 &amp;&amp;  echo -e "Network $hostname : ${GREEN}Online${NC}" || echo -e "${GRAY}Network $hostname${NC} : ${RED}Offline${NC}"  done &lt; list.txt         sleep 30 clear done   Is outputting info like this:    Network 10.x.xx.xxx : Online    Network 10.x.xx.xxx : Offline    Network 10.x.xx.xxx : Offline    Network 10.x.xx.xxx : Offline    Network 10.x.xx.x : Online    Network 139.xxx.x.x : Online    Network 208.xx.xxx.xxx : Online    Network 193.xxx.xxx.x : Online   which I'd like to clean up to get something like this:    Network 10.x.xx.xxx       : Online   Network 10.x.xx.xxx       : Offline    Network 10.x.xx.xxx       : Offline     Network 10.x.xx.x         : Online     Network 139.xxx.x.x       : Online   Network 208.xx.xxx.xxx    : Online     Network 193.xxx.xxx.x     : Online   Network 193.xxx.xxx.xxx   : Offline How do I find out which folder an app is in in iOS 11? In older versions of iOS, if I searched for an app with Spotlight, it would tell me which folder the app was in. (E.g., this feature still works on my iPad, which is running iOS 9, but not on my iPod Touch, which is running iOS 11.)  This feature seems to have been removed at some point. When I search for an app under iOS 11 using Spotlight, it will find the app and let me run it, but it doesn't tell me which folder it is in.  How can I find out which folder an app is in now? How to specify that search should be between two dates in technical specification I'm writing an integration manual that specifies how to use our WCF service for mobile messages and responses. The part in question needs to specify that the search results will be filtered between the start and end dates.  My aim is to make it clear that the time portion of the two dates will be discarded and the result will include everything from the start date to the end date. (In practice the code truncates the dates, adds one day to the end date, and returns the data that conforms to a query something like:  SELECT * FROM Table WHERE DateCreated >= @StartDate AND DateCreated &lt; @EndDate)  But my paragraph looks verbose and clumsy. It doesn't feel right. I feel kind of like John Cleese telling the kids when to put their clothes onto a lower peg. How can I make this less verbose?  Or maybe I'm over-thinking this?     When searching for responses, the search is performed between the supplied dates, after truncating the time portion thereof. The results will include responses with a date-time greater than equal to the start date, and less than one day after the end date. So passing the same value for start and end date will retrieve all responses received on that date, for messages previously sent by that user. Custom date format with jQuery validation plugin How can I specify a custom date formate to be validated with the  for jQuery? Can someone use DC generator along with electric motor to make electricity along with mechanical energy? I know this might sound stupid and maybe stupid totally, but I need some clarification....  I came through electric motor principle.  and then I came across the following link on DC generators    Now my question is, can't we combine them both to provide DC current with little DC used in motor side? By "combine," I mean using a DC generator coil only at the other side of the coil of the motor in the motor itself. The condition is that the 2 coils are connected via some apparatus that causes the 2nd coil to move along with the 1st coil.   In this way we can use both the mechanical energy produced by the 1st coil and the electrical energy produced by the 2nd coil, right?   This is a great concept if it works, right? Please throw some light on this. Changing my Apple ID's email: address is not available Equivalent of cURL for Ruby? How to use mysql JOIN without ON condition? Is the concept of even/odd numbers applicable to decimal numbers? For e.g. - 4.222 is a even number?