$\sum_{n=1}^\infty a(n)$ converges if and only if $\sum_{n=1}^\infty 2^na(2^n)$ converges
Proving Cauchy condensation test
I noticed something strange recently when trying to use rounded corners in pgfplots: it totally ruins the plot. See below without rounded corners:    and then with rounded corners:    Here is a minimal example:  \documentclass{standalone} \usepackage{pgfplots} \pgfplotsset{compat=newest}  \begin{document}  \begin{tikzpicture}[scale=0.8,trim axis left, trim axis right] \tikzset{every mark/.append style={scale=0.6}} \pgfplotsset{legend style={font=\footnotesize}}  \begin{axis}[   grid=both,             rounded corners,             ]     \addplot[samples=100,color=blue,very thick,smooth,domain=-10:10] {cos(x^(3))}; \end{axis} \end{tikzpicture}  \end{document}   Thanks in advance for your ideas ! What is the meaning of this sentence - "Were it not for the bodies there on the ground, it would have appeared to be a normal day at the oasis."? Possible Duplicate:          What is the meaning of this sentence - "Were it not for the bodies there on the ground, it would have appeared to be a normal day at the oasis."?  Does it mean something like this - If the bodies were not there on the ground, it would have appeared to be a normal day at the oasis.  Can anyone also explain the structure of this sentence and example of some similar sentences? We have two PHP5 objects and would like to merge the content of one into the second. There are no notion of subclasses between them so the solutions described in the following topic cannot apply.    //We have this: $objectA-&gt;a; $objectA-&gt;b; $objectB-&gt;c; $objectB-&gt;d;  //We want the easiest way to get: $objectC-&gt;a; $objectC-&gt;b; $objectC-&gt;c; $objectC-&gt;d;   Remarks:   These are objects, not classes. The objects contain quite a lot of fields so a foreach would be quite slow. So far we consider transforming objects A and B into arrays then merging them using array_merge() before re-transforming into an object but we can't say we are proud if this. Indefinite articles used with plural nouns: It was AN amazing TWO DAYS What are metaclasses in Python? Is “what’s” a correct short form of “what does”? E.g. “What’s he think?”  Usually ’s is short for “is” so I don’t know. $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$ (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$. Let $S$ be a bounded set. Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$. Note: If $\sup S$ is in $S$, it’s sufficient to define $s_n = \sup S$ for all $n$.  What I have tried so far:  Let $M=\sup S, L = \lim\, s_n,\,|s_n-L|&lt;\epsilon/2$, then $M-\epsilon &lt; L- \epsilon/2 &lt; s_n &lt; M &lt; M+\epsilon$. But I get this based on $M &lt;= \epsilon+L$, because $\epsilon+L$ is an upper bound for $s_n$.   However, as $s_n$ are just points in $S$, so I don't know if it's sufficient to say $M$, which is the $\sup$ of $S$, is smaller than $L+\epsilon$. Can EU citizens travel to UK after Brexit with just an identity card? sum of noncentral Chi-square random variables I need to find the distribution of the random variable  $$Y=\sum_{i=1}^{n}(X_i)^2$$ where $X_i\sim{\cal{N}}(\mu_i,\sigma^2_i)$ and all $X_i$s are independent. I know that it is possible to first find the product of all moment generating functions for $X_i$s, and then transform back to obtain $Y$'s distribution. However, I wonder whether there is a general form for $Y$ like the Gaussian case: we know the sum of independent Gaussian is still a Gaussian, and thus we only need to know the summed mean and summed variance.   How about all $\sigma^2_i=\sigma^2$? Will this condition make a general solution? Ephemeral error message: "cryptsetup: unknown fstype, bad password or options?" I activated disk encryption for my Ubuntu 15.10 64bit installation and get every time after entering the (correct) password the error message:  cryptsetup: unknown fstype, bad password or options?  But after few seconds (without any further interaction) this error message is replaced by the expected message:  cryptsetup: sda5_crypt set up successfully  That's it and the system is started.  Any suggestions what could be the problem? Is it merely cosmetic or serious? I am getting several new computers for my office.  I want to install Microsoft Office and my proprietary software and all updates on a single PC and create a system image that will be copied to every other PC.  How do I deal with Windows/Office product keys to prevent licence violations?  Am I able to do this from the System Image tool in Windows 7?  Note:  The PCs will very likely not be joined to my domain at this point if that makes a difference. I'm trying to do some scenes which use a mixture of 3D backgrounds with 2D animated characters.The best way I've figured this out so far is to use a plane with an image sequence of pngs which hold the frames.  The problem with this method is Blender seams to only load image sequences based on image numbering. If I wanted to do a sequence which loops say, frames 0-5, then repeats 4-0 for the 'return' of the animation, I would have to load frames doubled up.  Is there a way, maybe with a script, to change the sequence order? I can see it as a useful plugin but my python skills are pretty much non existent. How to insert text before the first line of a file? I've been looking around sed command to add text into a file in a specific line. This works adding text after line 1:  sed '1 a\   But I want to add it before line 1. It would be:  sed '0 a\   but I get this error: invalid usage of line address 0.  Any suggestion? What methods can be used to micro-adjust autofocus of a camera body to a particular lens? I have a Canon 7D with a 50mm f/1.4 lens, and I think the auto-focus of the lens is off.  How can I test and adjust this reliably?  Will this approach work with all of my lenses? If I had a different camera body, would I have other/different options? I know this sounds like a stupid question, but I just wanted to confirm if they are the same thing. In new canon, anti-gravity is never mentioned. How can I download an APK file from the Play Store? I can't download apps from the Play Store on my Android device (maybe due to a network issue), so I'm wondering how could I download the APK file of an app on my PC from ? How long typically are paper reviews? Is there such a thing as “too long”? When you review papers submitted for publication, is there an “optimal” length for reviews? In my experience as an author and referee, I have seen a large range of review lengths (for reference, a paper in my field is typically between 3 and 8 printed pages):   Zero length: for some of the papers I have authored, the reviewer just clicked the “accept as is” or “reject” checkmark on the review sheet, without adding any comment (at least, not any comment visible to me). It's not so common, but it has happened. Short length: a lot of time, I received reviews who consisted of a single paragraph. Medium: one full page, maybe two pages.   Although I have never received any such long reviews, I have myself written on a few occasions reviews that exceed two pages, including once or twice a four or five-page review.  These were cases where the paper was good, but could be (in my view) much improved and some of the aspects/consequences had escaped the authors' consideration.  I am wondering how useful reviews of various length are to editors. I often consider that “unmotivated” reviews are useless, as they do not give any real insight about the paper to the editor. For example, if the editor gets two conflicting non motivated reviews, how is she to decide?  On the other hand, although I write some from time to time, I have never received long reviews, so… is this something frowned upon? Find rational points on $x^2 + y^2 = 3$ and on $x^2 + y^2 = 17$ $(a)$ Find all rational points on the circle $x^2 + y^2 = 3$, if there are any. If there is none, prove so.  $(b)$ Find all rational points on the circle $x^2 + y^2 = 17$, if there are any. If there is none, prove so.  I'm not sure how proceed with finding a general formula (if there is one)  I know that for $(a)$ there is no rational points but I don't know how to explain that there are none.   whereas for $(b)$ there are such points, $(1,4)$ for example. I think that we can find the intersection between the line $y=m(x-1)+4$ and $x^2 + y^2 = 17$  Any help is appreciated! Prove that $x-1$ divides $x^n-1$ In algebra &amp; polynomials, how do we prove that $$x-1 \mid x^n -1?$$