if a series is convergent, prove the subsequence of its sequence is convergent.
Prove that the subsequence of a convergent series with nonnegative terms converges
Everyone knows that Wolverine's adamantium claws are super-sharp — sharper than even a razor — and virtually indestructible. What puzzled me the first time I saw Wolverine is: how do the claws stay in his body?  As it is shown, Wolverine's flesh and skin are human in strength, but the healing factor makes it superhumanly durable and strong. Still, when Wolverine slashes something, or someone, extremely hard, dense, or thick (as he has to use great amount of force), the claws don't tear through his hand and leave his hand through the opposite side of the palm (because of the action having equal and opposite reaction). if/then/else man page I would like to know if there is any man page documenting the construction of the most basic script commands like if/then/else, while, for each, and all the relative switches, like -eq, -e, -ge, and so on. Gapless superconductivity I don't understand what is meant by &quot;gapless superconductors&quot;. As far as I know, the superconducting gap is necessairy for both zero resistance and the Meissner effect, the two hallmarks of superconductivity. So, what does it mean to have gapless superconductivity and how does it work? Thanks in advance, Greetings How to trick a command into thinking its output is going to a terminal Given a command that changes its behaviour when its output is going to a terminal (e.g. produce coloured output), how can that output be redirected in a pipeline while preserving the changed behaviour?  There must be a utility for that, which I am not aware of.  Some commands, like grep --color=always, have option flags to force the behaviour, but the question is how to work around programs that rely solely on testing their output file descriptor.  If it matters, my shell is bash on Linux. I have a Asus ROG G551VW laptop and I'm trying to install Ubuntu 16.04 on my laptop, but when i try to load the installer, my laptop just freezes and its fan starts working for no reason. I have tried to install Ubuntu 14.04 on my laptop and the installation went smoothly. What should I do if I want to install Ubuntu 16.04 on my laptop?  Any help would be greatly appreciated. How to find all the tables in MySQL with specific column names in them? Calling PHP functions within HEREDOC strings In PHP, the HEREDOC string declarations are really useful for outputting a block of html.  You can have it parse in variables just by prefixing them with $, but for more complicated syntax (like $var[2][3]), you have to put your expression inside {} braces.  In PHP 5, it is possible to actually make function calls within {} braces inside a HEREDOC string, but you have to go through a bit of work.  The function name itself has to be stored in a variable, and you have to call it like it is a dynamically-named function.  For example:  $fn = 'testfunction'; function testfunction() { return 'ok'; } $string = &lt;&lt;&lt; heredoc plain text and now a function: {$fn()} heredoc;   As you can see, this is a bit more messy than just:  $string = &lt;&lt;&lt; heredoc plain text and now a function: {testfunction()} heredoc;   There are other ways besides the first code example, such as breaking out of the HEREDOC to call the function, or reversing the issue and doing something like:  ?&gt; &lt;!-- directly output html and only breaking into php for the function --&gt; plain text and now a function: &lt;?PHP print testfunction(); ?&gt;   The latter has the disadvantage that the output is directly put into the output stream (unless I'm using output buffering), which might not be what I want.  So, the essence of my question is: is there a more elegant way to approach this?  Edit based on responses: It certainly does seem like some kind of template engine would make my life much easier, but it would require me basically invert my usual PHP style.  Not that that's a bad thing, but it explains my inertia..  I'm up for figuring out ways to make life easier though, so I'm looking into templates now. How to find the sum of this series?  $$\sum_{k=0}^{\infty}\cfrac{{2}^{k}}{\binom{2k+1}{k}}$$  It seems very easy. But I still can not work it out, can anyone help? Which damage dice exactly does the Great Weapon Fighting fighting style allow you to reroll? Is there an online community to find travel partners (for mountains, sightseeing etc.)? Is there an online community to find travel partners for mountain trekking or sightseeing, or anything else?  I don't want to travel to foreign mountains alone, but I don't know where to search for company. There's a great community for accommodation (I mean CouchSurfing) so I think there should be something in "reverse" direction (searching not the place to stay but the people to go with). Is the function Riemann integrable? If yes, obtain the value of $\int_0^1f(x) \ dx$      $f(x) = \begin{cases} \frac {1}{n},  &amp; \frac{1}{n+1}&lt;x\le\frac{1}{n}\\ 0, &amp; x=0 \end{cases}$   My attempt  $f$ is bounded and monotonically increasing on $[0,1]$. Also, $f$ has infinite discontinuities but only one limit point. Therefore $f$ is Riemann integrable. Now, to calculate the integration  $\int_0^1f(x) \ dx=\int_{1/2}^{1}1 \ dx + \int_{1/3}^{1/2}\frac{1}{2} \ dx + \int_{1/4}^{1/3}\frac{1}{3} \ dx+...$  $=\sum_{n=1}^\infty \frac{1}{n^2}-\frac{1}{n}+\frac{1}{n+1}$  How do I proceed from here? How do I calculate these summations? I know $\sum \frac{1}{n}$ is $\log 2$, but not the other two summations. data & synchronization - manually sync mail. calender and contacts We can test the symmetry of a distribution around $0$ by Wilcoxon sign rank test, based on its sample.   But if we want to test if a distribution is symmetric around its mean, based on its sample $X_1, \dots, X_n$, is it valid to first normalize $X_i$ by the sample mean as $Y_i := X_i - \bar{X}$, and then apply Wilcoxon sign rank test to $Y_i$'s?  If not, what are some ways?  Thanks and regards! How to convert from one grading scheme to another? I'm trying to figure out the equivalence between grade schemes in different countries.  According to some Google results:   If you are a U.S. citizen, you need a GPA of 3.2+ (it varies, some say 3.2 and some others say 3.6, etc) to apply for a position in the UK (job or university) where the prerequisite is to have a minimum grade of 2:1. In UK 2:1 is earned if you have a 60-69% of the points. I've read that 2:1 is the same as 67%. In Spain we use a grading scheme of 10 points. So according to (2), for have a UK 2:1 you need to have 6-6.9 points. If you are Spanish and have 6.7, when converting it to GPA the result is 1.5.   However, if you take the above points and treat them as a math equation, the final result will be GPA 1.5 = GPA 3.2, which doesn't make any sense.  GPA 3.2 = UK 2:1 UK 2:1 = Spain 6.7 Spain 6.7 = GPA 1.5   Replacing now...  UK 2:1 = GPA 1.5     GPA 3.2 = GPA 1.5   My question is if someone has a better explanation on converting grades obtained in different countries. What is an "AppImage"? How do I install it? I just read . It talks about downloading an AppImage.     Yes, Linux; the Linux packages is distributed as an .appimage for 32-bit and 64-bit distributions, and should run across all major Linux distributions without any issues. The team currently have no plan to provide a native .deb (or .rpm) installer.   What are AppImages? How do they differ from snaps? structs in C with initial values Using the fact that $n^3 + 5n$ is divisible by $6$, infer that $n^4 - 2n^3 + 11n^2 + 14n$ is always divisible by $24$ The problem is as stated in the title. I've tried factorization, but it doesn't seem to yield anything. Any help is greatly appreciated. Also please don't use modular arithmetic, thank you. Speed of light travel I have just started special theory of relativity. The limiting speed known as speed of light fascinated me most. I asked my teacher:      Consider two massless objects moving in the same direction at the speed of light. What would be their relative velocity w.r.t each other?   My teacher tells me that their relative velocity would be zero. But then the speed of light is universal constant regardless of the motion of its frame of reference, so shouldn't their relative velocity be $c$? What is their relative velocity and how? Do objects moving at the speed of light obey law of addition of velocities? What is the difference between an abstract method and a virtual method? What is the difference between an abstract method and a virtual method? In which cases is it recommended to use abstract or virtual methods? Which one is the best approach? What, precisely, is a confidence interval?