Does the $\sum^{\infty}_{j=1}(b_j)^3$ converge if $\sum^{\infty}_{j=1}(b_j)$ converges?
If $\sum_{1}^{\infty}(a_n)^3$ diverges, does $\sum_{1}^{\infty}(a_n)$?
There's a spaceship, nearing the end of its journey. Everybody on board is in cryosleep. For some reason the wake-up system malfunctions, and the ship will fly on indefinitely. The people at the space port realise this, and try to wake someone up. Their last resort is finding someone on board who is telepathic, and wake this person up. They find one, but it turns out to be an unborn baby. Ordeal of Heliod and changing creature control Here is the situation that came up:  Player 1 controls a creature enchanted with  and two +1/+1 counters. On Player 2's turn, they cast  to take control of the creature, and then attack with it. The attack triggers the Ordeal, putting another +1/+1 counter on it. Now that it has three counters, it is sacrificed and a player gains 10 life.  Which player sacrifices the enchantment and gains 10 life? The controller of the creature (Player 2) or the player who cast the enchantment (Player 1)? Ping is a a program owned by root with the user id bit set.  $ ls -l `which ping` -rwsr-xr-x 1 root root 35752 Nov  4  2011 /bin/ping   As I understand it, if a user runs the ping process, then the effective user id will change from the real user id (i.e. the user id of the person who launched the process) to the user id root. However when I try this and look at the output of ps to see if the ping process is running as the root user, I still get the real user id showing.  ps -e -o user,ruser,euser,cmd,args | grep ping sashan   sashan   sashan   ping -i 10 -c 1000 www.goog ping -i 10 -c 1000 www.google.com The question asks to prove that -  $$\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x}  = \frac 12 (\tan 27x - \tan x) $$  I tried combining the first two or the last two fractions on the L.H.S to allow me to use the double angle formula and get $\sin 6x$ or $\sin 18x$ but that did not help at all.   I'm pretty sure that if I express everything in terms of $x$, the answer will ultimately appear but I'm also certain that there must be another simpler way. We've never had an explicit policy around downvoting -- users are free to, if they like, cast all their votes as downvotes. Whether this behavior is desirable or not is another matter.   Now, we do discourage downvoting by making downvotes cost -1 rep to the casting voter. But, there's nothing in our policy that says continual downvoting is bad or wrong or inappropriate.  However, my thoughts on this are changing. I no longer think it is community friendly behavior to cast an extreme number of downvotes, and I am considering an explicit policy forbidding it.  Some data points (as of today):  Total SO upvotes cast: 45,555,530  Total downvotes cast: 365,192  users with more than 100 downvotes cast: 596  users with > 100 downvotes and the lowest ratio of upvotes to downvotes:           |           |         | up:down user id | downvotes | upvotes |  ratio --------+-----------+---------+---------  187712 |       201 |      37 |    0.18    2525 |       580 |     149 |    0.25  136829 |       137 |      43 |    0.31   55943 |       438 |     140 |    0.31   19405 |       236 |      94 |    0.39    5640 |      3299 |    1337 |    0.40  180659 |       126 |      66 |    0.52   78613 |       348 |     198 |    0.56   39892 |       181 |     109 |    0.60     432 |       310 |     197 |    0.63  238704 |       198 |     132 |    0.66   12643 |       170 |     116 |    0.68   18256 |       219 |     152 |    0.69  122428 |       167 |     118 |    0.70  118854 |       120 |      86 |    0.71  105662 |       273 |     201 |    0.73  187690 |       135 |     101 |    0.74    1782 |       573 |     430 |    0.75   (this is a complete list, there are no more below this threshold.)  Based on this data I am leaning towards enforcing a "no more downvotes may be cast" if    You have at least (n) votes cast total Your upvote to downvote ratio is lower than ~ 0.5 I don't want to use the Super for that, I want to use Alt+Space instead. "Scaled $L^p$ norm" and geometric mean The $L^p$ norm in $\mathbb{R}^n$ is \begin{align} \|x\|_p = \left(\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} Playing around with WolframAlpha, I noticed that, if we define the "scaled" $L^p$ norm in $\mathbb{R}^n$ to be  \begin{align} \| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p} \end{align} then \begin{align} \lim_{p \to 0} \|x\|_p &amp;= \left( \prod_{j=1}^{n} |x_j| \right)^{1/n}, \end{align} which is the geometric mean of the coordinates' absolute values. This is interesting maybe because the $L^p$ norm doesn't have a nice limit at zero.  My questions:   How do I prove this? Is this definition of "scaled $L^p$ norm" interesting, or known by another name, or used anywhere? Is there any interesting reason to define the $L^0$ norm as the geometric mean, as above? Further reading?   Thanks! Could anyone provide an example of a nonempty subset $U$ of $R^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $R^2$ I updated to TeXLive 2017 and couldn't get newlfm.cls to work without errors.  Here is a minimal example:  \documentclass{newlfm} \begin{document} \begin{newlfm} \end{newlfm} \end{document}   LaTex is throwing undefined control sequence errors on \fancy@reset, \f@ncyolh, and \f@ncyorh.  What causes the errors?  What am I missing? Newsletter open and click tracking use http. Can i configure it that it uses https with our certificate? Can this cause block bounces? The event horizon of a black hole is where gravity is such that not even light can escape. This is also the point I understand that according to Einstein time dilation will be infinite for a far-away-observer.  If this is the case how can anything ever fall into a black hole.  In my thought experiment I am in a spaceship with a powerful telescope that can detect light at a wide range of wavelengths.  I have it focused on the black hole and watch as a large rock approaches the event horizon.  Am I correct in saying that from my far-away-position the rock would freeze outside the event horizon and would never pass it? If this is the case how can a black hole ever consume any material, let alone grow to millions of solar masses.  If I was able to train the telescope onto the black hole for millions of years would I still see the rock at the edge of the event horizon?  I am getting ready for the response of the object would slowly fade.  Why would it slowly fade and if it would how long would this fading take?  If it is going to red shift at some point would the red shifting not slow down to a standstill? This question has been bugging me for years!  OK - just an edit based on responses so far.  Again, please keep thinking from an observers point of view.  If observers see objects slowly fade and slowly disappear as they approach the event horizon would that mean that over time the event horizon would be "lumpy" with objects invisible, but not passed through?  We should be able to detect the "lumpiness" should we not through? How to get color of pixel in coordinates 123, 456 in screen? My AppleScript app needs to get color of pixel in coordinates 123, 456 in screen. Why does this test program result in a java.lang.IllegalMonitorStateException?  public class test {     static Integer foo = new Integer(1);     public static void main(String[] args) {         synchronized(foo) {             foo++;             foo.notifyAll();         }         System.err.println("Success");     } }   Result:  Exception in thread "main" java.lang.IllegalMonitorStateException         at java.lang.Object.notifyAll(Native Method)         at test.main(test.java:6) Let $\alpha(t)$ be a regular curve. Suppose there is a point $a$ in $R^3$ such that $\alpha(t)-a$ is orthogonal to $T(t)$ for all t. Prove that $\alpha(t)$ lies on a sphere.  So, I let $\alpha(t)=(x(t), y(t),z(t))$ and $a=(a,b,c)$.  Since, $\alpha(t)-a$ is orthogonal to $T(t)$,  $&lt;\alpha(t)-a,T(t)&gt;=0$.  But I don't know how to solve this.. A classic exponential inequality: $x^y+y^x>1$ Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides.    My Attempt: Let the trapezium be $OABC$ and that the O is a origin and the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$.Then the equation of $OB$ diagonal is $\vec{r}=\vec{0}+\lambda \vec{b}................(1)$ And equation of $AC$ diagonal is $\vec{r}=\vec{a}+\mu(\vec{c}-\vec{a}).......(2)$ And the equation of the line joining the mid points of $OA$ i.e.$\frac{\vec{a}}{2}$ and $BC$ i.e. $\frac{\vec{b}+\vec{c}}{2}$ is $\vec{r}=\frac{\vec{a}}{2}+t(\frac{\vec{b}+\vec{c}}{2}-\frac{\vec{a}}{2}).........(3)$. Here $\lambda,\mu,t$ are scalars. I do not know how to solve $(1)$ and $(2)$ and put into $(3)$ to prove the desired result. Please help me.Thanks. Issue in using longtable within table environment I have an issue making a long list with wrapped text. The table can be several page long. I use longtable and assign the width for each column but as soon as I put it in the \begin{table} \end{table}, it does not work. It clips the table. I want to make a label for the table so I put it in the table environment.  And is there a way to automatically adjust the width so that I don't need to assign the value manually?  Thank you.   Tung  \begin{table} \centering \begin{longtable}{|p{6.5cm}|p{2.5cm}|p{2.5cm}|p{2.5cm}|} \hline Name                                      &amp; Company     &amp;  Catalog Number   &amp;   Comments    \\ \hline Small DNA FRAG Extract Kit-100PR          &amp; VWR         &amp;   97060-558 &amp; \\ \hline Acrylamide 40\% solution 500 mL           &amp; VWR         &amp;   97064-522 &amp;  \\ \hline Bis-acrylamide 2\% (w/v) solution 500 mL  &amp; VWR         &amp;   97063-948 &amp;  \\ \hline GeneRuler 100 bp DNA Ladder, 100-1,000 bp &amp; Fermentas   &amp;   SM0241  &amp;    \\ \hline Mini Vertical PAGE System                 &amp; VWR         &amp;   89032-300  &amp;  \\ \hline \end{longtable} \label{tab:list} \end{table} Many anime do not follow the manga history and are made shorter. However there are a few that do follow it or try to finish the history with the manga like the big three (Naruto, Bleach, One Piece), Death Note, and FMA Brotherhood. But others finish the anime without the real ending like FMA, Ouran High School, Pandora Hearts, Tsubasa Reservoir Chronicle. Why is this so? Is it because of the high cost? Or the lack of popularity? In the  library, they are overloading the () operator.  Is this a convention in C++? For callbacks, etc.?  I have seen this in code of a co-worker (who happens to be a big Boost fan). Of all the Boost goodness out there, this has only led to confusion for me.  Any insight as to the reason for this overload? Prove the following inequality:   $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$   This should be solved using Taylor series.  I tried expanding the left to the 5th degree and the right site to the 3rd degree, but it didn't help.  Any tips?